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In this thesis we consider some identities in rings having some additional structures.
There are three subjects:

(1) Special identities with (a,b)-derivations.
Let R be a prime ring. In 1993 Breˇsar studied the identity f1(x) f2(y) =f3(x) f4(y) for all x,y 2 R where each fi is a derivation of R. In 1997 Chang considered a more general case when f2 and f3 are (a,b)-derivations, f1 is an (a,a)-derivation and f4 is a (b,b)-derivation. We consider a more general case when each fi is an (ai,bi)-derivation. We show that there exists an
invertible element t in symmetric Martindale ring of quotients of R such that
f1(x) = f3(x)t and f4(x) = t f2(x) for all x 2 R.

(2) On graded polynomial identities with an antiautomorphism.
Let G be a commutative monoid with cancellation and let R be a strongly
G-graded associative algebra with finite G-grading and with an antiautomorphism.
Suppose that R satisfies a graded polynomial identity with an antiautomorphism.
We show that R is a PI algebra.

(3) Posner’s theorems for superderivations on superalgebras.
Let A = A0 A1 be a graded-prime associative superalgebra over a commutative
associative ring F with 1
2 , and let Zs(A) be its supercenter. If d is a
superderivation of A such that [x,d(x)]s 2 Zs(A) for all x 2 A, then either A
is commutative, or d(A0) = 0 and d(A1) Z(A), in particular, d(A) = 0 if
|d| = 0. On superalgebras, the composition of two nonzero superderivations
may still be a superderivation.

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