In abstract algebra , an ideal is a subset of an algebraic structure with at least one multiplicative two-digit operation that is closed with respect to products with elements from the entire structure.
The ideals of the same type on a given algebraic structure always form a system of envelopes called the ideal system . For every ideal system there is always a corresponding envelope operator (and vice versa), that is the corresponding ideal operator .
For the sake of simplicity, only the commutative case is described here. If one waives the commutativity of the multiplication, then the following are left ideals , and if the left and right factors are swapped for each product , right ideals result accordingly . Bilateral ideals, or just ideals, are both left and right ideals. In commutativity, there is no difference between these three types of ideals.
Ring ideals
Number-theoretical investigations of number ranges , in which a clear prime factorization of elements was no longer given, led to the development of the "classical" ideal theory for commutative rings .
definition
If a ring, then a ( Dedekind's ) ideal or ideal is the carrier set of a subgroup of for which:
![(R, +, \ cdot)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7a0f4d832c9b7871f68bc77313edbd25f82717e)
![A_ {d} \ subseteq R](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9ff4ba4891017ad2910dccc30fa954bc4a4237e)
![(R, +)](https://wikimedia.org/api/rest_v1/media/math/render/svg/466356a631ac93bc70fbe2d276117d22f980e285)
![\ forall r \ in R \; \ forall a \ in A_ {d}:](https://wikimedia.org/api/rest_v1/media/math/render/svg/044a442bf53d11c9d546e9237d51b82610c32d5f)
![r \ cdot a \ in A_ {d}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/446146f9307c2fccbbc9d77a63c4284f6d1d0e42)
properties
- The ideals of a ring are precisely the kernels of the ring homomorphisms of the ring.
- The ideals of a ring each form an envelope system , so that the ideals are given by the associated envelope operator .
![(\;) _ {d}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ed0f1d9b881cd16fbc4aec199737e2e2044b841)
Remarks
General ideal operators
Since, as a rule, only the respective associative two-digit operation is decisive for the factorization (the non-associative case is not dealt with below), it is sufficient for a general ideal theory to consider semigroups:
In the following, a commutative multiplicative semigroup is always given , and it is
![(S, \ cdot)](https://wikimedia.org/api/rest_v1/media/math/render/svg/a67d66f9afab5e95334226e6183e85a25d1c37ba)
![\ cdot: {\ mathfrak P} (S) \ times {\ mathfrak P} (S) \ to {\ mathfrak P} (S), (A, B) \ mapsto A \ cdot B: = \ {a \ cdot b \ mid a \ in A, b \ in B \},](https://wikimedia.org/api/rest_v1/media/math/render/svg/910afdf4a43938422b741cad1b939e8732ae365f)
the complex multiplication over , where is the power set of .
![(S, \ cdot)](https://wikimedia.org/api/rest_v1/media/math/render/svg/a67d66f9afab5e95334226e6183e85a25d1c37ba)
![{\ mathfrak P} (S): = \ {A \ mid A \ subseteq S \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59a61b90f7e08303da03e7f5c183cbe7032b3993)
![S.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
then forms a commutative, associative, complete multiplicative lattice with a zero element .
![\ emptyset](https://wikimedia.org/api/rest_v1/media/math/render/svg/6af50205f42bb2ec3c666b7b847d2c7f96e464c7)
definition
It should now
![(\;) _ {{x ^ {*}}}: {\ mathfrak P} (S) \ to {\ mathfrak P} (S), A \ mapsto (A) _ {{x ^ {*}}} ,](https://wikimedia.org/api/rest_v1/media/math/render/svg/9769f3dc199969cb1946427a688942d14010ed81)
be an envelope operator on , with the property that
![S.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
![\ forall A \ in {\ mathfrak P} (S):](https://wikimedia.org/api/rest_v1/media/math/render/svg/602adbb27591592cbcc4adc151f52b21a19d1702)
![S \ cdot (A) _ {{x ^ {*}}} \ subseteq (A) _ {{x ^ {*}}}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb2bdca1149d3fc28fbd77a8c8de5cb74cd30406)
Then, a -Idealoperator or shortly operator on called, which is -Idealsystem or system to a means -ideal and is of produced -Great . denotes of generated -ideal and is of generated - principal ideal .
![x ^ {*}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5be23ee5d433f8b576e63bcb47518128ee0b6bb)
![(S, \ cdot)](https://wikimedia.org/api/rest_v1/media/math/render/svg/a67d66f9afab5e95334226e6183e85a25d1c37ba)
![{\ mathfrak I} _ {{x ^ {*}}}: = \ {(A) _ {{x ^ {*}}} \ mid A \ in {\ mathfrak P} (S) \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71b11f5a75d043aedbfb760bb07e6d81f20c9392)
![x ^ {*}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5be23ee5d433f8b576e63bcb47518128ee0b6bb)
![(\;) _ {{x ^ {*}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d29e0996951cb22bd6055229d2b735ce18f3275)
![A _ {{x ^ {*}}} \ in {\ mathfrak I} _ {{x ^ {*}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9566380e7e1f7f29d48847ea3019ce224a4dea80)
![x ^ {*}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5be23ee5d433f8b576e63bcb47518128ee0b6bb)
![(A) _ {{x ^ {*}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee127c431cff3fdc6b074204883ec92a0c42ba33)
![A \ in {\ mathfrak P} (S)](https://wikimedia.org/api/rest_v1/media/math/render/svg/37e88a66406629adc20d792d0e6ac4ca87ceec9f)
![x ^ {*}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5be23ee5d433f8b576e63bcb47518128ee0b6bb)
![(a_ {1}, \ cdots, a_ {n}) _ {{x ^ {*}}}: = (\ {a_ {1}, \ cdots, a_ {n} \}) _ {{x ^ { *}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5ee2a422d4b89c7a4b0d9b29206243187c174f5)
![{\ displaystyle a_ {1}, \ cdots, a_ {n} \ in S, n \ in \ mathbb {N},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9862f015531d625614bf00523a6c0ef483128af9)
![x ^ {*}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5be23ee5d433f8b576e63bcb47518128ee0b6bb)
![(a) _ {{x ^ {*}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af35afbd7ee7eec234fcad255c175e057357142e)
![a \ in S](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe43ae205afd3f10432bddabc3bdae5ecfa6b412)
![x ^ {*}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5be23ee5d433f8b576e63bcb47518128ee0b6bb)
comment
-
is usually not an ideal, but because it is advantageous for ideal arithmetic, it should also be a spurious main ideal here , if .![(\ emptyset) _ {{x ^ {*}}} = \ bigcap {\ mathfrak I} _ {{x ^ {*}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/548ae5327193cc09c98b74d6127757e9bf733e5d)
![x ^ {*}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5be23ee5d433f8b576e63bcb47518128ee0b6bb)
![(\ emptyset) _ {{x ^ {*}}} = \ emptyset](https://wikimedia.org/api/rest_v1/media/math/render/svg/f55eb4bfb08759ff8fa63cf9347dfc8be2ae95e9)
- To distinguish between ideals and any subsets of , the ideals, in contrast to any subsets, are given a corresponding index in the following.
![S.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
Ideal associations
Up are two two-digit operations
![{\ mathfrak I} _ {{x ^ {*}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd9a046ab7034728c171350f34a955df85037ca5)
![\ vee _ {{x ^ {*}}}: {\ mathfrak I} _ {{x ^ {*}}} \ times {\ mathfrak I} _ {{x ^ {*}}} \ to {\ mathfrak I} _ {{x ^ {*}}}, (A _ {{x ^ {*}}}, B _ {{x ^ {*}}}) \ mapsto A _ {{x ^ {*}}} \ vee _ {{x ^ {*}}} B _ {{x ^ {*}}}: = (A _ {{x ^ {*}}} \ cup B _ {{x ^ {*}}}) _ {{x ^ {*}}},](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc680a272b6be7d58309a74a30114cebb78a8721)
![\ wedge _ {{x ^ {*}}}: {\ mathfrak I} _ {{x ^ {*}}} \ times {\ mathfrak I} _ {{x ^ {*}}} \ to {\ mathfrak I} _ {{x ^ {*}}}, (A _ {{x ^ {*}}}, B _ {{x ^ {*}}}) \ mapsto A _ {{x ^ {*}}} \ wedge _ {{x ^ {*}}} B _ {{x ^ {*}}}: = (A _ {{x ^ {*}}} \ cap B _ {{x ^ {*}}}) _ {{x ^ {*}}},](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d3f5c0c237b3f1a6b84f2cbc11190634b3549b8)
given so that it forms a complete association , the association of the ideals of . It is the -Idealverbindung , the -Idealdurchschnitt .
![({\ mathfrak I} _ {{x ^ {*}}}, \ vee _ {{x ^ {*}}}, \ wedge _ {{x ^ {*}}})](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a9d1f2676d33c6cf4082fc63271ec5a01098278)
![x ^ {*}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5be23ee5d433f8b576e63bcb47518128ee0b6bb)
![(S, \ cdot)](https://wikimedia.org/api/rest_v1/media/math/render/svg/a67d66f9afab5e95334226e6183e85a25d1c37ba)
![\ vee _ {{x ^ {*}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/784bc41dc46dcf0439f21a118acc626e38b1af39)
![x ^ {*}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5be23ee5d433f8b576e63bcb47518128ee0b6bb)
![\ wedge _ {{x ^ {*}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd6a5f89e8d6b442753668139d8558532baea7b)
![x ^ {*}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5be23ee5d433f8b576e63bcb47518128ee0b6bb)
As for all envelope systems, the following also applies to every ideal system:
![x ^ {*}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5be23ee5d433f8b576e63bcb47518128ee0b6bb)
![\ forall A, B \ in {\ mathfrak P} (S):](https://wikimedia.org/api/rest_v1/media/math/render/svg/f206e8be11d184b38f953551f9876c15d864c0fb)
![A _ {{x ^ {*}}} \ wedge _ {{x ^ {*}}} B _ {{x ^ {*}}} = A _ {{x ^ {*}}} \ cap B _ {{x ^ {*}}}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/e389f0d8eaf34f98732f36e56f15ac1b38e56a0a)
Algebraic Idea Operators
is algebraic if and only if is algebraic , that is
![\ forall a \ in S \; \ forall A \ in {\ mathfrak P} (S):](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a2d62b86485ba752e31ec9ee7ae92aac9001e82)
-
and
Denotes the power of the crowd , then exists with
![| A |](https://wikimedia.org/api/rest_v1/media/math/render/svg/648fce92f29d925f04d39244ccfe435320dfc6de)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![{\ displaystyle (\;) _ {x_ {s} ^ {*}}: {\ mathfrak {P}} (S) \ to {\ mathfrak {P}} (S), A \ mapsto (A) _ { x_ {s} ^ {*}}: = \ bigcup \ {(N) _ {x ^ {*}} \ mid N \ subseteq A, | N | \ in \ mathbb {N} _ {0} \}, }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4fa298cbc2d6c3c6f80f7c7dbd2e1a0c03ee74c)
always an algebraic -idea operator to it
.
![x ^ {*}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5be23ee5d433f8b576e63bcb47518128ee0b6bb)
![(\;) _ {{x ^ {*}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d29e0996951cb22bd6055229d2b735ce18f3275)
x ideal operators
The ideal multiplication
![\ cdot _ {{x ^ {*}}}: {\ mathfrak I} _ {{x ^ {*}}} \ times {\ mathfrak I} _ {{x ^ {*}}} \ to {\ mathfrak I} _ {{x ^ {*}}}, (A _ {{x ^ {*}}}, B _ {{x ^ {*}}}) \ mapsto A _ {{x ^ {*}}} \ cdot _ {{x ^ {*}}} B _ {{x ^ {*}}}: = (A _ {{x ^ {*}}} \ cdot B _ {{x ^ {*}}}) _ {{x ^ {*}}},](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5a8b374de2a7e54378c7773b9ddb469a62069f5)
has the property characteristic of ideals
![\ forall A _ {{x ^ {*}}}, B _ {{x ^ {*}}} \ in {\ mathfrak I} _ {{x ^ {*}}}:](https://wikimedia.org/api/rest_v1/media/math/render/svg/eaf35df6a33dbbda6b91f3ac64d0782075e18d76)
![B _ {{x ^ {*}}} \ cdot _ {{x ^ {*}}} A _ {{x ^ {*}}} \ subseteq A _ {{x ^ {*}}},](https://wikimedia.org/api/rest_v1/media/math/render/svg/91895683dc5da0dcec32528dd6f4756a1963d3ba)
In general, however, it does not yet offer enough properties to be able to investigate properly. On the other hand, the following class of ideal operators has proven to be well suited for a general ideal theory .
![(S, \ cdot)](https://wikimedia.org/api/rest_v1/media/math/render/svg/a67d66f9afab5e95334226e6183e85a25d1c37ba)
![x ^ {*}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5be23ee5d433f8b576e63bcb47518128ee0b6bb)
definition
So-called -Idea operators or -operators are -Idea operators in which
translations![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
![(\;) _ {x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c9e3ab4ed89120f4d411f478d2eb4a56b5d6b9e)
![\ forall t \ in S:](https://wikimedia.org/api/rest_v1/media/math/render/svg/880fd840a536ac09803543965b2b93d7ca705b72)
![{\ displaystyle \ vartheta _ {t} \ colon S \ to S, a \ mapsto \ vartheta _ {t} (a): = a \ cdot t,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46e5013612aa5c25d2c4b4710c8f9271c5dfc6c4)
As with topological closure operators, " continuous " are :
![\ forall t \ in S \; \ forall A \ in {\ mathfrak P} (S):](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3b2234e2f9d0aca0e1016ca2080b61e24195812)
![\ vartheta _ {{t}} {\ bigl (} (A) _ {x} {\ bigr)} \ subseteq {\ bigl (} \ vartheta _ {{t}} (A) {\ bigr)} _ { x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/991a94be2e8c919be0079ea7f915394933ffe6a9)
with for each and everyone .
![\ vartheta _ {{t}} (A): = \ {\ vartheta _ {{t}} (a) \ mid a \ in A \} = A \ cdot \ {t \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21c3ea679dd73ffb829feebd783771c246b4bd7a)
![t \ in S](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4e4de5cbd0086a33eceb5150ae9c19a73dde4be)
![A \ in {\ mathfrak P} (S)](https://wikimedia.org/api/rest_v1/media/math/render/svg/37e88a66406629adc20d792d0e6ac4ca87ceec9f)
properties
- With every -idea operator there is also an -idea operator.
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
![(\;) _ {x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c9e3ab4ed89120f4d411f478d2eb4a56b5d6b9e)
![(\;) _ {{x_ {s}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d09cee07994537aa1783dd963e1b5689ff2f9c9b)
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
- For every -ideal operator on even follows
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
![(\;) _ {x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c9e3ab4ed89120f4d411f478d2eb4a56b5d6b9e)
![(S, \ cdot)](https://wikimedia.org/api/rest_v1/media/math/render/svg/a67d66f9afab5e95334226e6183e85a25d1c37ba)
![\ forall A, B \ in {\ mathfrak P} (S):](https://wikimedia.org/api/rest_v1/media/math/render/svg/f206e8be11d184b38f953551f9876c15d864c0fb)
![(A) _ {x} \ times B \ subseteq (A \ times B) _ {x}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/98e5da3aaf8ff4dbf5177c61707301e0ffd49494)
- The two-sided ideals of a semigroup are exactly the kernels of certain semigroup homomorphisms of , and it holds
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
![(S, \ cdot)](https://wikimedia.org/api/rest_v1/media/math/render/svg/a67d66f9afab5e95334226e6183e85a25d1c37ba)
![(S, \ cdot)](https://wikimedia.org/api/rest_v1/media/math/render/svg/a67d66f9afab5e95334226e6183e85a25d1c37ba)
![\ forall A, B \ in {\ mathfrak P} (S):](https://wikimedia.org/api/rest_v1/media/math/render/svg/f206e8be11d184b38f953551f9876c15d864c0fb)
![(A) _ {x} \ cdot _ {x} (B) _ {x} = (A \ cdot B) _ {x}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f5bf7ff8d18bcd1aac49cb4581891d4cc936263)
- A two-sided ideal system forms a (commutative,) associative, quasi-logical and complete multiplicative association .
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
![({\ mathfrak I} _ {x}, \ vee _ {x}, \ wedge _ {x}, \ cdot _ {x})](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8703b281e1e633b706e2374b4a951bff6eb1826)
- Likewise, for two-sided ideals there is such a multiplicative lattice, which is also always algebraic.
![({\ mathfrak I} _ {{x_ {s}}}, \ vee _ {{x_ {s}}}, \ wedge _ {{x_ {s}}}, \ cdot _ {{x_ {s}} })](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef42979d89ff45b1487c990db46364871bd3d6a4)
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
Remarks
- Any -ideal operator always induces an -ideal operator, so that -ideal operators are also of a very general nature.
![x ^ {*}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5be23ee5d433f8b576e63bcb47518128ee0b6bb)
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
- Another, more abstract approach to a general ideal theory is the description of ideal systems using appropriate multiplicative associations.
- As a rule, terms from “classical” ideal theory, such as maximum ideal , prime ideal , etc., can be used for -ideals without any problems .
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
r -Ideal operators
definition
An -Ideaoperator auf
is an -Idealoperator, which is also
translation closed , so
![(\;) _ {r}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b8020da3db5fb37b844440b6ed6662889602fe4)
![(S, \ cdot)](https://wikimedia.org/api/rest_v1/media/math/render/svg/a67d66f9afab5e95334226e6183e85a25d1c37ba)
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
![\ forall t \ in S \; \ forall A_ {r} \ in {\ mathfrak I} _ {r}:](https://wikimedia.org/api/rest_v1/media/math/render/svg/0af0fa2c00fb3cd94121f6c64a81f859ba3d39db)
![\ vartheta _ {{t}} (A_ {r}) \ in {\ mathfrak I} _ {r},](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e7e472beecfc378973eb87a1525149b07986378)
and for which also applies:
![\ forall a \ in S:](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb8be86f8f044554c3cc375e1559170208093b36)
![(a) _ {r} = \ {a \} \ cup \ vartheta _ {{a}} (S).](https://wikimedia.org/api/rest_v1/media/math/render/svg/36162d0642cf950b01d3cc3edaaebb570d784d3d)
properties
- For every translation-completed -ideal operator on even follows
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
![(\;) _ {x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c9e3ab4ed89120f4d411f478d2eb4a56b5d6b9e)
![(S, \ cdot)](https://wikimedia.org/api/rest_v1/media/math/render/svg/a67d66f9afab5e95334226e6183e85a25d1c37ba)
![\ forall t \ in S \; \ forall A \ in {\ mathfrak P} (S):](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3b2234e2f9d0aca0e1016ca2080b61e24195812)
![\ vartheta _ {{t}} {\ bigl (} (A) _ {x} {\ bigr)} = {\ bigl (} \ vartheta _ {{t}} (A) {\ bigr)} _ {x }.](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb5b944e5928ec89713e05495d650028875dd0d1)
- If a unit has 1, then every translation-completed -idea operator is already an -idea operator and
![(S, \ cdot)](https://wikimedia.org/api/rest_v1/media/math/render/svg/a67d66f9afab5e95334226e6183e85a25d1c37ba)
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
![(\;) _ {x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c9e3ab4ed89120f4d411f478d2eb4a56b5d6b9e)
![(S, \ cdot)](https://wikimedia.org/api/rest_v1/media/math/render/svg/a67d66f9afab5e95334226e6183e85a25d1c37ba)
![r](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538)
![\ forall a \ in S:](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb8be86f8f044554c3cc375e1559170208093b36)
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and
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is also an ideal operator.![r](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538)
- Every two-sided main ideal is a multiplication ideal , that is
![r](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538)
![\ forall a \ in S \; \ forall A_ {r} \ in {\ mathfrak I} _ {r}:](https://wikimedia.org/api/rest_v1/media/math/render/svg/1cba33c6255da03f821fc1d26716310b6ed800c9)
![A_ {r} \ subset (a) _ {r} \ iff \ exists B_ {r} \ in {\ mathfrak I} _ {r}: B_ {r} \ cdot _ {r} (a) _ {r} = A_ {r} \ neq (a) _ {r}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce3f853036066857eb09cc30020eea06aa32808f)
- A two-sided one can be shortened , so
![(a) _ {r}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d2aa88b520892afceee650f28227391c62fa6d6)
![\ forall a \ in S \; \ forall A_ {r}, B_ {r} \ in {\ mathfrak I} _ {r}:](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c02426ad6f0cc3dd6abbba1ef3125f4813fe490)
![A_ {r} \ cdot _ {r} (a) _ {r} = B_ {r} \ cdot _ {r} (a) _ {r} \ implies A_ {r} = B_ {r},](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8c495a51d9f1eefa5c9c604b34c8cd7369f60d3)
- if in can be shortened .
![a \ in S](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe43ae205afd3f10432bddabc3bdae5ecfa6b412)
comment
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Ideal systems have all the essential properties of the ideal systems of rings, which is why they allow a good investigation of the divisibility relationships in .![d](https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab)
![(S, \ cdot)](https://wikimedia.org/api/rest_v1/media/math/render/svg/a67d66f9afab5e95334226e6183e85a25d1c37ba)
literature
- H. Examiner: Investigations into the divisibility properties of bodies . In: J. reine angew. Math. Band 168 , 1932, pp. 1-36 .
- K. E. Aubert: Theory of x-ideals . In: Acta Math. Band 107 , 1962, pp. 1-52 .
- I. Fleischer: Equivalence of x-systems and m-lattices . In: Colloquia Mathematica Societatis Janos Bolyai . 33. Contributions to Lattice Theory, Szeged, 1980. North Holland , Amsterdam / Oxford / New York 1983, pp. 381-400 .
- P. Lorenzen: Abstract foundation of the multiplicative ideal theory . In: Math. Z. Band 45 , 1939, pp. 533-553 .
- M. Ward, R. P. Dilworth: The lattice theory of ova . In: Ann. Math. Band 40 , 1939, pp. 600-608 .
- L. Fuchs :: Partly ordered algebraic structures . Vandenhoeck & Ruprecht , Göttingen 1966.
- G. Birkhoff: Lattice Theory . 3. Edition. American Mathematical Society , Providence (R. I.) 1973.