The adjunct of a single element is used in mathematics when one wants to embed a ring without a single element in a ring with a single element, for example in order to be able to apply a theorem that only applies to rings with a single element.
Rings
Be any ring. Then define the operations
on the Cartesian product
![{\ displaystyle A \ times \ mathbb {Z}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab0a51e6b0276a3a869fc52ffbfb3c7b1e6d04f6)
![{\ displaystyle (a, \ lambda) + (b, \ mu) \, = \, (a + b, \ lambda + \ mu)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d70b648d0bed96b710f524c2cc4d4390d9ccce52)
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,
whereby . Note that products can be formed using the obvious - module structure. Simple calculations show that with these operations there is a ring with the one element . If one identifies with, one can understand an element as writing and as a subring of . The above definitions are then written in the following expected form:
![{\ displaystyle a, b \ in A; \, \ lambda, \ mu \ in \ mathbb {Z}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/863ee02bbc924fbe14d845f4d32f18e6508a6e26)
![{\ displaystyle \ lambda b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4421a4aedebbd90361980e61938c6a11e608acd4)
![\ mathbb {Z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc)
![{\ displaystyle A_ {1}: = A \ times \ mathbb {Z}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f30d77371440f753cc1057f8acc25af97f97725)
![{\ displaystyle e: = (0.1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad27908febfddb56a11874d956eba374938df7d4)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![{\ displaystyle A \ times \ {0 \} \ subset A \ times \ mathbb {Z},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ce613bd1a36f1abdd7a12add9d6273b069dbd59)
![{\ displaystyle (a, \ lambda)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b26e9c5a3c98ea416c1a27af2b8511b99c28b577)
![{\ displaystyle a + \ lambda e}](https://wikimedia.org/api/rest_v1/media/math/render/svg/941b32c6a95c1d3f6827676d710bf69c04013f5f)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![A_ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc2435b217c1a0f46f8a517ffa225c6f9440e81)
![{\ displaystyle a + \ lambda e \, + \, b + \ mu e \, = \, a + b + (\ lambda + \ mu) e}](https://wikimedia.org/api/rest_v1/media/math/render/svg/384d9bcc002b5ff01adf18737df9021c3c7821b0)
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.
This means that each ring can be embedded in a ring with a single element. If you already had a one element, you get a new one, the original one of is no longer a one and the characteristic of is 0, even if it had a positive characteristic.
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![A_ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc2435b217c1a0f46f8a517ffa225c6f9440e81)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![{\ displaystyle A_ {1},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eaac19dd9a383457a52222837be415c8b1354909)
![A_ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc2435b217c1a0f46f8a517ffa225c6f9440e81)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
In the above construction, a two-sided ideal is in and it applies . Since there is no zero divisor , even a prime ideal is in .
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![A_ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc2435b217c1a0f46f8a517ffa225c6f9440e81)
![{\ displaystyle A_ {1} / A \ cong \ mathbb {Z}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ecdab7c19a54cdcc7ed3d6040b225b1db0b464b)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![A_ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc2435b217c1a0f46f8a517ffa225c6f9440e81)
Algebras
If there is not only a ring but even an algebra over a solid , the above construction can be adapted so that the resulting ring is again an algebra. This one has merely by replacing, which means you then forms . The -algebra structure is given by the formula
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![\ mathbb {Z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![{\ displaystyle A_ {1}: = A \ oplus K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0919052d1741dbaecddcf9ff24e54c3e4bc169d2)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![{\ displaystyle \ mu \ cdot (a + \ lambda e): = \ mu a + \ mu \ lambda e}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1850f4ab799612127ed40dac6dfa2d672f4af35f)
given. When the adjunct of a unit is mentioned in the context of algebras, this construction is usually meant. Again, a two-sided ideal is in and it holds . Since there is a body, there is even a maximal ideal in .
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![A_ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc2435b217c1a0f46f8a517ffa225c6f9440e81)
![{\ displaystyle A_ {1} / A \ cong K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d404917f1b5f488b6f7ccadd380de865d4fb4dba)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![A_ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc2435b217c1a0f46f8a517ffa225c6f9440e81)
Normalized algebras
If a standardized algebra or even a Banach algebra is above , where for or stands, one can also make a standardized algebra in which one
![{\ displaystyle (A, \ | \ cdot \ |)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/033f1e38ad07346a224c1b766587fe518c19f8cf)
![\ mathbb {K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1848c435e64864e9ad4efa7e46bd6bc900c35c99)
![\ mathbb {K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1848c435e64864e9ad4efa7e46bd6bc900c35c99)
![\ mathbb {R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc)
![\ mathbb {C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7)
![A_ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc2435b217c1a0f46f8a517ffa225c6f9440e81)
![\ mathbb {K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1848c435e64864e9ad4efa7e46bd6bc900c35c99)
![{\ displaystyle \ | a + \ lambda e \ |: = \ | a \ | + | \ lambda |}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2ec45d8201670c5c4b52f16bfd0dda7bb2ddd0d)
puts. That certainly makes a normalized space, and the multiplicative triangular inequality of carries over to , because
![A_ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc2435b217c1a0f46f8a517ffa225c6f9440e81)
![{\ displaystyle (A, \ | \ cdot \ |)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/033f1e38ad07346a224c1b766587fe518c19f8cf)
![{\ displaystyle (A_ {1}, \ | \ cdot \ |)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35b76058656e2678010d675df65d72c64d5abae3)
= : = = = .
![{\ displaystyle \ | ab + \ lambda b + \ mu a + \ lambda \ mu e \ |}](https://wikimedia.org/api/rest_v1/media/math/render/svg/122136c6c67b07e10326ba765fb4172414aaa1c7)
![{\ displaystyle \ | ab + \ lambda b + \ mu a \ | + | \ lambda \ mu | \ leq \ | a \ | \ | b \ | + | \ lambda | \ | b \ | + | \ mu | \ | a \ | + | \ lambda || \ mu |}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cd8eef0a63b0bf538d735cf625816cfa05a8410)
![{\ displaystyle (\ | a \ | + | \ lambda |) (\ | b \ | + | \ mu |)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/512af825594a983431404bcf99608d48786ac14c)
![{\ displaystyle \ | a + \ lambda e \ | \ cdot \ | b + \ mu e \ |}](https://wikimedia.org/api/rest_v1/media/math/render/svg/adb43d493fe222bcd4c248742f71db36e738f7b0)
Is a Banach algebra, that is, as normed space completely , it is also a Banach algebra.
![A_ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc2435b217c1a0f46f8a517ffa225c6f9440e81)
If there is a -Banach algebra with involution , the involution can be given by the formula
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![{\ displaystyle a \ mapsto a ^ {*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/160bff67bd6d02365007df4eac1a382a4bb0d3bf)
![{\ displaystyle (a + \ lambda e) ^ {*}: = a ^ {*} + {\ overline {\ lambda}} e}](https://wikimedia.org/api/rest_v1/media/math/render/svg/478ac59ed07bda84a99422e3622c77b3a41d126c)
to expand. If the involution is isometric , the same is true for .
![A_ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc2435b217c1a0f46f8a517ffa225c6f9440e81)
![A_ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc2435b217c1a0f46f8a517ffa225c6f9440e81)
C * algebras
If a C * -algebra has no unit element, the above construction does not produce a C * -algebra . But you can have a different standard to choose the algebra also makes a C *. To do this, one sets
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![A_ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc2435b217c1a0f46f8a517ffa225c6f9440e81)
![A_ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc2435b217c1a0f46f8a517ffa225c6f9440e81)
![A_ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc2435b217c1a0f46f8a517ffa225c6f9440e81)
-
.
This is precisely the operator norm for left-hand multiplication .
![{\ displaystyle L_ {a + \ lambda e}: A \ rightarrow A, b \ mapsto (a + \ lambda e) b = ab + \ lambda b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86e3011c894bec5b4f138ac364308d504a1f188d)
swell
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Jacques Dixmier : Les C * -algèbres et leurs représentations (Les grands classiques Gauthier-Villars). Éditions Gabay, Paris 1996, ISBN 2-87647-013-6 (unchanged reprint of the Paris 1969 edition)
- Louis H. Rowen: Ring Theory, Vol. 1 (Pure and applied mathematics; Vol. 127). Academic Press, Boston, Mass. 1988, ISBN 0-12-599841-4 .