Narrow Results

Let R be a ring and σ an endomorphism of R. We recall that R is called an (S, 1)-ring if for a, b ∈ R, ab = 0 implies aRb = 0. We involve σ to generalize this notion. We say that R is a left σ-(S, 1) ring if for a, b ∈ R, ab = 0 implies aRb = 0 and σ(a)Rb = 0. We say that R is a right σ-(S, 1) ring if for a, b ∈ R, ab = 0 implies aRb = 0 and aRσ(b) = 0. R is called a σ-(S, 1) ring if it is both right and left σ-(S, 1) ring. In this paper we give examples of such rings and a relation between σ-(S, 1) rings, 2-primal rings, and σ(⁎)-rings.

We show that a certain class of matrix rings, with suitable endomorphisms σ are left σ-(S, 1) but not right σ-(S, 1), and vice versa.

We show an application of Hochschild cohomology to the moduli of subalgebras of the full matrix ring without proofs. We also calculate Hochschild cohomology Hⁱ(S₁₁(R), M₃(R)/S₁₁(R)) for an R-subalgebra S₁₁(R) of M₃(R) over a commutative ring R. The calculation by using a spectral sequence will give us a useful technique for calculating Hochschild cohomology.