Let A be a finite dimensional algebra over a field F. Let R and S be biregular algebras over F such that 1R ∈ R and 1S ∈ S. We show that if R/P≃A≃ S/M for each primitive ideal P in A and each primitive ideal M in S then End FR≃ End S implies R≃S.

A ring R is called an l-ring (r-ring) in case R contains an indentity and every left (right) semigroup ideal is a left (right) ring ideal. A number of structure theorems are obtained for l-rings when R is left noetherian and left artinian. It is shown that left noetherian l-rings are local left principal ideal rings. When R is a finite dimensional algebra over a field, the property of being an l-ring is equivalent to being an r-ring. However, examples are given to show that these two concepts are in general not equivalent even in the artinian case.

We show that rings for which every non-constant multiplicative endomorphism is additive are trivial or power rings (that is, rings R such that R = R2 and x2 = 0 = x+x for all x ∈ R) and that if R is a power ring for which every multiplicative endomorphism is additive, then End (R) is a zero semigroup or a semilattice according to how the product is defined.

A semigroup over a generalized tree, denoted by the term ℳL-semigroup, is a compact semigroup S such that Green's relation H is a congruence on S and S/H is an abelian generalized tree with idempotent endpoints and E(S/H) a Lawson semilattice. Each such semigroup is characterized as being constructible from cylindrical subsemigroups of S and the generalized tree S/H in a manner similar to the construction of semigroups over trees and of the hormos. Indeed, semigroups over trees are shown to be particular examples of the construction given herein.

A sandwich semigroup of continuous functions consists of continuous functions with domains all in some space X and ranges all in some space Y with multiplication defined by fg = foαog where α is a fixed continuous function from a subspace of Y into X. These semigroups include, as special cases, a number of semigroups previously studied by various people. In this paper, we characterize the regular elements of such semigroups and we completely determine Green's relations for the regular elements. We also determine the maximal subgroups and, finally, we apply some of these results to semigroups of Boolean ring homomorphisms.