On the algebraic closure in rings
The "algebraic closure" of a subset K C A of a ring is an algebraic analogue of topological closure.
Operators consistent in regularity
If S(X) is an arbitrary subset of L(X) (where L(X) is the set of all bounded operators on a Banach space X), then we say that B ∈ L(X) is S-consistent, or consistent in S(X), provided that for all A ∈ L(X) the following holds: AB ∈ S(X) if and only if BA ∈ S(X). It is convenient to take that S(X) is close to the set of all invertible operators on X, or that S(X) contains regular operators. Here “regular” means that S(X) is equal to the set of invertible, left (right) invertible, Fredholm, left (right) Fredholm, Weyl, or Browder operators on X. In this article we completely describe operators consistent in the previous regularities.
AXIOMATISABILITY OF WEAKLY FLAT, FLAT, AND PROJECTIVE S-ACTS
ABSTRACT This paper gives necessary and sufficient conditions on a monoid S that the class of flat left S-acts be (first order) axiomatisable. It presents also a streamlined version of the result of Stepanova that the class of projective left S-acts is axiomatisable if and only if the strongly flat left S-acts are axiomatisable and S is left perfect.