| Abstract: | ABSTRACT Let K denote a commutative ring with unity and A be a K-algebra. An element, d ∈ A is said to be left self distributive, or LSD, if dxy = dx dy for all x, y ∈ A. Let ?(A) be the set of LSD elements. Similarly, one can define the set of right self distributive, or RSD, elements and let ?(A) be the set of RSD elements. Let 𝒟(A) = ?(A) ∩ ?(A), the set of self distributive, or SD, elements. An algebra, A, is said to be left self distributively generated, or LSD-generated, if A = mod K (?(A)), the K-module generated by ?(A). Analogously, one defines RSD-generated and SD-generated algebras. If A = mod K (?(A)) = mod K (?(A)), then A is said to be LSD/RSD-generated, which is a strictly larger class than the class of SD-generated algebras. Examples are given to illustrate the variety of LSD-generated algebras. This paper continues the study of LSD-generated, RSD-generated, LSD/RSD-generated and SD-generated algebras. This paper characterizes exactly which structural matrix rings are LSD-generated. The paper begins with an important lemma that characterizes LSD elements in a matrix ring in terms of the entries of the matrix. The main result characterizes those structural matrix rings that are LSD-generated, first in terms of a 2 × 2 generalized matrix ring, then strictly in terms of the shape of the matrix ring. Sharper results are obtained for LSD/RSD-generated and SD-generated structural matrix rings. The final section is devoted to an application of this result to endomorphism rings. If the endomorphism ring of a finitely generated module is a homomorphic image of a structural matrix ring, then the module is a direct sum of cyclic modules. Further conditions are given to describe when the structural matrix ring is LSD-generated, in terms of the annihilators of the generating set. |
