A module [ moːdʊl ] (masculine, plural modules [ moːdʊln ], the declination is similar to that of consul ; of Latin modulus , diminutive of mode , "degree", "unit") is an algebraic structure that a generalization of a vector space represents.
Similar to rings , a module is understood to mean something different depending on the sub-area and textbook. The definitions of morphisms as well as substructures and superstructures are also slightly different. In mathematical terms, these different module terms are different categories .
Modules over a commutative ring with a single element
A module over a commutative ring or module for short is an additive Abelian group together with a mapping
![(R, +, \ cdot)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7a0f4d832c9b7871f68bc77313edbd25f82717e)
![(M, +)](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac85e31c15b188cc6674c500245dd710fc78d8ca)
-
(called multiplication by scalars, scalar multiplication),
so that:
![r_ {1} \ cdot (r_ {2} \ cdot m) = (r_ {1} \ cdot r_ {2}) \ cdot m](https://wikimedia.org/api/rest_v1/media/math/render/svg/e779698812cced3038de46045721708a8cdc16a2)
![(r_ {1} + r_ {2}) \ times m = r_ {1} \ times m + r_ {2} \ times m](https://wikimedia.org/api/rest_v1/media/math/render/svg/746a77caa5950aacecbb2a9cc7ddc6756c767b2c)
![r \ cdot (m_ {1} + m_ {2}) = r \ cdot m_ {1} + r \ cdot m_ {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02468e237193fd19c87d00848bd23b1d5b69aa45)
If one requires additionally for an identity with
![(R, +, \ cdot)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7a0f4d832c9b7871f68bc77313edbd25f82717e)
![1](https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf)
-
,
so one calls the module unitary (English: unital ). Some authors generally require the existence of a unit element for rings, and then likewise for modules over rings. If there is a body , i.e. also forms an Abelian group, then the unitary modules are over just the vector spaces over .
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![{\ displaystyle (R \ backslash \ {0_ {R} \}, \, \ cdot)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a50371861f304494e71cdf0bbff654262be2e0ad)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
Comment: The concept of vector space is actually superfluous, since it is a special case of the more general concept of unitary module. In fact, however, the additional condition that there is a body enables so many results that are not correct in the general situation that it is customary to separate the special case from the general case with a separate term.
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
The study of modules on commutative rings is the subject of commutative algebra .
Abelian groups
Each additive Abelian group is uniquely a unitary module; H. a unitary module over the commutative ring of integers. Be . Because of
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![\ mathbb {Z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc)
![m \ in G](https://wikimedia.org/api/rest_v1/media/math/render/svg/977f3b62e05c59d1cf1ca26fd5980426bdb48c2b)
![{\ displaystyle 1 \ cdot m = m, \, 0 \ cdot m = {\ mathfrak {0}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c3f2733e152012caff0853f0d2b102db252c4fc)
must apply to with :
![k \ in \ Z](https://wikimedia.org/api/rest_v1/media/math/render/svg/59a12237af5f2ec5fc7c5023f439266bae1380f7)
![k \ geq 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/79214ef55efadfb1d9c9b02252eb8a71cf6f8b6b)
![{\ displaystyle k \ cdot m = \ underbrace {(1+ \ dotsb +1)} _ {k {\ text {times}}} \ cdot m = \ underbrace {m + \ dotsb + m} _ {k {\ text {times}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89956534ddd8352292df1e57e91db9993b4a452c)
and analog:
-
Since this only possible connection fulfills the modulus axioms, the claim follows. The following number ranges are additive groups and thus modules:
![\ mathbb {Z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc)
Upper rings as modules
Is a top ring of , it is by definition an Abelian group.
![(S, +, \ cdot)](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b3b36b3cc3d5b82cd67519ccf4f2d42715f6ea1)
![(R, +, \ cdot)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7a0f4d832c9b7871f68bc77313edbd25f82717e)
![{\ displaystyle (S, +)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/757e19e5bf3fd2aa1fcc68e2ff18d9805acc2f9c)
If the ring multiplication of is restricted to the set , then this defines the necessary scalar multiplication in order to consider it as a module over in a natural way . If and if they have the same single element, the module is unitary.
![S.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
![R \ times S](https://wikimedia.org/api/rest_v1/media/math/render/svg/55c3ff61b6cca09ae2b3fb47ba9417b51d83b94e)
![S.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![S.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
If and even bodies are, one speaks in this situation of a body expansion . The module structure then becomes a vector space structure as described above. The consideration of this vector space structure is an indispensable aid in the investigation of body extensions.
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![S.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
Note: The number ranges mentioned in the previous chapter are all upper rings of , which also shows that they are naturally modules.
![\ mathbb {Z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc)
![\ mathbb {Z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc)
Vector spaces with a linear map in themselves
Let be the polynomial ring over a body . Then, the corresponding -modules one-to-one pairs consisting of a vector space and an endomorphism on .
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![K [X]](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bb4d802ca5718a14dc961af8692f35cdfad169b)
![(V, A)](https://wikimedia.org/api/rest_v1/media/math/render/svg/340fd05a90d7cb89118cfd953ce1f3a7bddd43ad)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![V](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
- Be a module. We note that a vector space is as in embedded. Let this vector space. The pair that belongs to it is now , being through
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![K [X]](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bb4d802ca5718a14dc961af8692f35cdfad169b)
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![K [X]](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bb4d802ca5718a14dc961af8692f35cdfad169b)
![V](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![(V, A)](https://wikimedia.org/api/rest_v1/media/math/render/svg/340fd05a90d7cb89118cfd953ce1f3a7bddd43ad)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![V \ to V, \ quad v \ mapsto X \ cdot v.](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d01543bed3a3e487db798e5d772fdcd36316c73)
- given is.
- We define a module structure for a pair
![(V, A)](https://wikimedia.org/api/rest_v1/media/math/render/svg/340fd05a90d7cb89118cfd953ce1f3a7bddd43ad)
![K [X]](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bb4d802ca5718a14dc961af8692f35cdfad169b)
![{\ displaystyle X \ cdot v: = A (v)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/509ad5a0e7736792e22866ebe7a58da3c4912189)
- and continue the -linear on , d. H. for all
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![K [X]](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bb4d802ca5718a14dc961af8692f35cdfad169b)
![p (X) = a_ {0} + a_ {1} X + a_ {2} X ^ {2} + \ dotsb + a_ {n} X ^ {n} \ in K [X]](https://wikimedia.org/api/rest_v1/media/math/render/svg/16463716fcac1d29b5e8b42d9e23e6c91b3d7a3c)
- let's set
-
.
Ring ideals
Each ring is a module on itself with ring multiplication as an operation. The sub-modules then correspond exactly to the ideals of (since in this section is commutative, we do not need to distinguish between left and right ideals).
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
Modules over any ring
Let it be a ring . If this ring is not (necessarily) commutative, a distinction must be made between left and right modules.
![(R, +, \ cdot)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7a0f4d832c9b7871f68bc77313edbd25f82717e)
A - left module is an Abelian group along with a ring and a map
![(M, +)](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac85e31c15b188cc6674c500245dd710fc78d8ca)
![(R, +, \ cdot)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7a0f4d832c9b7871f68bc77313edbd25f82717e)
![R \ times M \ to M, \ quad (r, m) \ mapsto r \ cdot m = rm,](https://wikimedia.org/api/rest_v1/media/math/render/svg/abece0bb0715a27829488765fd9ee2e9e2ea4cc6)
which is additive in both arguments, d. H. for all true
![r, r_ {1}, r_ {2} \ in R, \, m, m_ {1}, m_ {2} \ in M](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa52f2c46abb1b2fc335e51cd79a0b4840613ec0)
-
and
![r \ cdot (m_ {1} + m_ {2}) = r \ cdot m_ {1} + r \ cdot m_ {2},](https://wikimedia.org/api/rest_v1/media/math/render/svg/0420a122fb2204f0913a60555bb8d98312d64ac6)
and for the
-
for all
applies.
If it is assumed that there is a unitary ring with a one-element , then it is usually also required that the left module is unitary (English: unital ), i.e. H.
![(R, +, \ cdot)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7a0f4d832c9b7871f68bc77313edbd25f82717e)
![1](https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
-
for everyone .![m \ in M](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c5790f9afa538086b9fea356114e77099c1a775)
Some authors generally require the existence of a single element for rings and modules.
A right module is defined similarly, except that the scalars of the ring act from the right on the elements of :
A - right module is an Abelian group together with a mapping that is additive in both arguments
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![M \ times R \ to M, \ quad (m, r) \ mapsto m \ cdot r = mr,](https://wikimedia.org/api/rest_v1/media/math/render/svg/20a34f2aeb3cb8961af7d0738598d6903ed6fab7)
so that
-
for all
A right module over a unitary ring with one element is unitary if
![1](https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf)
-
applies to all .![m \ in M](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c5790f9afa538086b9fea356114e77099c1a775)
If commutative , the terms left and right module match (except for the spelling), and one speaks simply of - modules . Usually the above notation is used for left modules.
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
Alternative definitions
- A left module is an Abelian group together with a (possibly unitary) ring homomorphism
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![R \ to {\ mathrm {End}} _ {\ mathbb {Z}} (M).](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7804888a2e6245d1aabf61acb1ea3a6a29592fc)
- It is the ring of endomorphism of the concatenation as a product:
![{\ mathrm {End}} _ {\ mathbb {Z}} (M)](https://wikimedia.org/api/rest_v1/media/math/render/svg/c88c4fd90729e1900295f81cd6426fa99683e949)
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
-
For
- A right module is an Abelian group together with a (possibly unitary) ring homomorphism
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![R \ to ({\ mathrm {End}} _ {\ mathbb {Z}} (M)) ^ {{\ mathrm {op}}}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/d91cfbca1cc269d4dc900a2c57bbdbd74c67de5b)
- It should be the counter-ring of endomorphism ring, that is the ring of endomorphisms of the legal chaining as a product:
![({\ mathrm {End}} _ {\ mathbb {Z}} (M)) ^ {{\ mathrm {op}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2573d461cdcb8756f352abb99fc6f8db98dd4537)
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
-
For
Bimodules
Let it be and rings. Then a - -Bimodule is an Abelian group together with a -Left module- and a -Right module structure, so that
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![S.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![S.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![S.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
-
For
applies.
For unitary rings and a unitary - -Bimodule (i.e. with for all ) can alternatively be described as an Abelian group together with a unitary ring homomorphism
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![S.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![S.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
![{\ displaystyle 1_ {R} \ cdot m = m \ cdot 1_ {S} = m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88ffdc42894638363b0a483200df5693a2fdc141)
![m \ in M](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c5790f9afa538086b9fea356114e77099c1a775)
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![{\ displaystyle R \ otimes _ {\ mathbb {Z}} S ^ {\ mathrm {op}} \ to \ mathrm {End} _ {\ mathbb {Z}} \, M.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/334f6d886f0eaff5f6cf42e9e5456ce47a0b4034)
That means: A unitary - -Bimodule is nothing more than a unitary -Link module.
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![S.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
![{\ displaystyle R \ otimes _ {\ mathbb {Z}} S ^ {\ mathrm {op}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dae43fb264b184e92883e8f29001cad34912c014)
Change of ring
and be rings and be a ring homomorphism. The regulation defines
for each module![S.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
![\ rho \ colon S \ to R](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f1a73c4cb35429ebf546acf5a28fc5030df14b2)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![(s, m) \ mapsto \ rho (s) m](https://wikimedia.org/api/rest_v1/media/math/render/svg/0df4ad48b05b6052593ee2eecde52e3d4cd3ae7d)
a module structure called the one associated with and the module structure . This module is labeled with or with . If in particular a subring of and is the canonical embedding, then the module obtained by restricting the scalars from to is called.
![S.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![\ rho](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![S.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
![\ rho _ {*} (M)](https://wikimedia.org/api/rest_v1/media/math/render/svg/1878779dd008fe123a4e56d59bc1b3ff9d298ad2)
![M _ {{[S]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cef63fa722b81d5e5a248b65a058e3518f94f377)
![S.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![\ rho](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64)
![\ rho _ {*} (M)](https://wikimedia.org/api/rest_v1/media/math/render/svg/1878779dd008fe123a4e56d59bc1b3ff9d298ad2)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![S.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
![S.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
If a sub-module is of , then is a sub-module of and![N](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3)
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![\ rho _ {*} (N)](https://wikimedia.org/api/rest_v1/media/math/render/svg/a59b314eb81b9e485dc9618e7f894ae32e1d453b)
![\ rho _ {*} (M)](https://wikimedia.org/api/rest_v1/media/math/render/svg/1878779dd008fe123a4e56d59bc1b3ff9d298ad2)
Modules over an associative algebra
If a commutative ring and an associative R -algebra , then a - left module is a module together with a module homomorphism
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![A \ otimes _ {R} M \ to M, \ quad a \ otimes m \ mapsto am,](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3de480be099c00196619d012611ef1a9a3c71b9)
so that
-
For
applies.
A - Law Module is a module together with a -Modulhomomorphismus
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![M \ otimes _ {R} A \ to M, \ quad m \ otimes a \ mapsto ma,](https://wikimedia.org/api/rest_v1/media/math/render/svg/d611b72da04b4d8cefdd01a29d9eec64c7cbc746)
so that
-
For
applies.
Unitary modules and bimodules are defined analogously to the case of rings.
Modules over a Lie algebra
Let it be a Lie algebra over a field . A - module or a representation of a vector space together with a -bilinearen Figure
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![{\ mathfrak {g}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28)
![{\ mathfrak {g}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![\ mathfrak g \ times M \ to M, \; (X, m) \ mapsto X \ cdot m,](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d520d0a12bb44c15e250ba8582dd241158f7264)
so that
-
For
applies.
Alternatively, a module is a vector space along with a homomorphism of Lie algebras over![{\ mathfrak {g}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![{\ mathfrak g} \ to {\ mathfrak {gl}} (M);](https://wikimedia.org/api/rest_v1/media/math/render/svg/26b3086ed02d9101ef6d6b0e815543d748b16a7c)
where is the -algebra of the endomorphisms of with the commutator as a Lie bracket.
![{\ mathfrak {gl}} (M)](https://wikimedia.org/api/rest_v1/media/math/render/svg/a407f10f176b0eb866d454eb5f49129ce6387e0e)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
-Modules are the same as modules under the universal enveloping algebra of .
![{\ mathfrak {g}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28)
Modules over a group
It is a group . A module or, more precisely, left module is an Abelian group together with an outer two-digit link![(G,*)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e54a87abf331634c8962ef14c4c5ec41f94fd29c)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
-
,
so that
-
for all
and
-
for all
such as
-
for the neutral element of and for everyone![e](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
applies.
A - law module is defined similarly; the second condition is through
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
-
for all
to replace.
Alternatively, a - (left) module is an Abelian group together with a group homomorphism
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![(M, +)](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac85e31c15b188cc6674c500245dd710fc78d8ca)
![G \ to {\ mathrm {Aut}} _ {\ mathbb {Z}} (M),](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9be3f53c615fd9427e8d00dd8c31ac5fb50bdec)
here is the group of automorphisms from with the link
![{\ mathrm {Aut}} _ {\ mathbb {Z}} (M) = ({\ mathrm {End}} _ {\ mathbb {Z}} (M)) ^ {\ times}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2839f4ca097c8e18e7acf21937ce6b788739fb69)
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
-
For
A right module is an Abelian group together with a group homomorphism
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![(M, +)](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac85e31c15b188cc6674c500245dd710fc78d8ca)
![G \ to ({\ mathrm {Aut}} _ {\ mathbb {Z}} (M)) ^ {{\ mathrm {op}}},](https://wikimedia.org/api/rest_v1/media/math/render/svg/f36dc35dbcae03ae43fd7aedf12ba6c8a8323eb9)
the product on is through
![({\ mathrm {Aut}} _ {\ mathbb {Z}} (M)) ^ {{\ mathrm {op}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04ec02461efc0c877d05be9ed06d35737f4cef1b)
-
For
given.
If further is a ring, then a - module is an Abelian group with a module structure and a module structure which are compatible in the following sense:
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
-
For
Alternatively, a - module is a module together with a group homomorphism
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![G \ to {\ mathrm {Aut}} _ {R} (M),](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac6ce1a0e12562711e284d5dc475b8ebc6e63ad8)
this is the group of automorphisms of the module.
![G \ to {\ mathrm {Aut}} _ {R} (M)](https://wikimedia.org/api/rest_v1/media/math/render/svg/c099eb22c607f65a8b06b643aef3ad41c1c3c0a2)
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
- modules are the same as modules above the group ring .
![R [G]](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cfc4570ea0faa6453735827f1e877904ecf78fe)
If a body is special , the concept of the - module corresponds to that of the linear representation of .
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
See also
Web links
Alexander von Felbert: Introduction to module theory .
literature
References and comments
-
↑ not to be confused with scalar product
-
↑ Here the Abelian group was written additively.
-
↑ Such a module does not have to have a basis , namely for modules with torsion elements .
-
^ Nicolas Bourbaki: Elements of Mathematics, Algebra I, Chapters 1–3 . 2nd Edition. Springer, 1998, ISBN 3-540-64243-9 , § 3. Tensor products , 2., p. 221 ( Internet Archive ).