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

By Konstant J. Beidar

"Discusses the newest effects in regards to the zone of noncommutative ring concept often called the speculation of generalized identities (GIs)--detailing Kharchenko's effects on GIs in best earrings, Chuang's extension to antiautomorphisms, and using the Beidar-Mikhalev thought of orthogonal of entirety within the semiprime case. offers novel proofs of current results."

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3)Now let 4 3 ( 4 = (VY)(34 [Ilv # 011 *{ l l w # Oll AllzYz = 011 }]. For an element P E R,R I= 43 (P) if and only if the right annihilator of P in R is an essential right ideal of R. Two formulas q5 = q5(yl, y2, . . ,yn) and II,= @(zl,q , . . ,zm) are said to be equivalent, if for any R-A-ring R and any substitution q E Mup(X,R), the formula +(x:, . , is true in R if and only if $(z:, . , z L ) is true. The following proposition is easily proved by induction on complexity of formulas. ,$x) 42 CHAPTER 1.

T of T , where each is either A1 or A2, and then extend by linearity. The mapx : TI@T2@.. @Tm-+ P given by t l @ t 2 @ . , @ t , ~ t ~ t(~ w h. e. r. etp~= a o r p = T d e p e n d i n g on whether ti E A1 or ti E Az) is already K-linear, and from the nature of multiplication in T it is easy to see that $0 is a K-algebra homomorphism. 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.

PRELIMINARIES 10 Proof. Using property (ii)' the map x : V @ . . @ V +P given by v1 @ . . @ vn H ~ ? v $ .. V: is a well-defined Klinear map, and extension of x by additivity to K{V} yields the required K-algebra map 11,. 2 Let a be an automorphism of K, let P be a K-algebra, and let 6 : V + P be a a-semilinear map. Then: ( a ) q5 can be uniquely extended to a a-homomorphism of Kalgebras $ : K{V} + P. (b) 4 can be uniquely extended to a a-antihomomorphism of K-algebras $ : K{V} + P. Proof. To prove (a) we consider P as a K-algebra P* by defining k x = k'x, k E K, x E P.

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