Rings with Generalized Identities (Pure and Applied by Konstant J. Beidar

By Konstant J. Beidar

"Discusses the most recent effects about the sector of noncommutative ring idea often called the idea of generalized identities (GIs)--detailing Kharchenko's effects on GIs in best jewelry, Chuang's extension to antiautomorphisms, and using the Beidar-Mikhalev conception of orthogonal final touch within the semiprime case. offers novel proofs of present results."

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That R(RA,~(ABC)) A partial ordering on theset Y = S < X > is said to be a semigroup ordering if B < B’ with B,B’ E Y implies ABC < AB’C for all A, C E Y. A partial ordering 5 on the set S < X > is said to be compatible with A if for any o E A the element f, is a linear combination of monomials V with v < W,. Denote by I = I(A) the two-sided ideal of @ generated by the elements W, - f,, o E A. Clearly the @-module I is generated by the elements A(W,,- &)B,where A , B E Y ,o E A. 1,. we denote by O(h) E E(Y) the set of all maximal monomials in h E @.

V, E R we set Y = {yl, y2,. vl;$2,: : . ,v,) E independent of q. ,vn] if q5(v1, v2, . . ,v), = 1. In this case we wili say tlidk Eke formula $(v1, v2,. . ,v,) is true in the a-A-ring k. Otli&wise we will say that the formula q5(w1, 212,. . ,W,) is false Examples. Let R be a ring. = yzll. Then R q51 if and only if R is commutative. (2) Let q52(z) ,= (b'y)llzy = yzll: ,Given any P E R, R $ 2 ( ~ ) if and only if P is a central element of R. (3)Now let 4 3 ( 4 = (VY)(34 [Ilv # 011 *{ l l w # Oll AllzYz = 011 }].

2) it is clear that $0 maps I to 0. As a result $0 may by lifted to a K-algebra homomorphism 4 : T/I + P by defining (94 = t40, -E = t I , t E T. Thecommutativity of the above diagram then yields the commutativity of + which shows that property (ii) holds. The existence and uniqueness of a coproduct of AI and A2 having been established, wenow refer to the coproduct of A1 and A2 and denote it by A1 AS. In general A;" n A[ may properly contain K. For instance, the reader may check that Q Q provides such an example.

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