Regularity of semilattice sums of rings

on direct sums of baer modules

the notion of baer modules was defined recently

Subdirect Sums of Rings

On Regularity of Rings

It is shown that a ring R is (von Neumann) regular if and only if every cyclic left R-module is GP -injective if and only if R is left PP and left GP -injective. In addition, two examples are given to answer two open questions in the negative. 2000 Mathematics Subject Classification: 16E50, 16D60

Artinian Band Sums of Rings

Band sums of associative rings were introduced by Weissglass in 1973. The main theorem claims that the support of every Artinian band sum of rings is finite. This result is analogous to the well-known theorem on Artinian semigroup rings. 1991 Mathematics subject classification (Amer. Math. Soc): primary 16P20, 16W50; secondary 20M25. Let B be a band, that is, a semigroup consisting of idempoten...

Sums of Squares in Real Analytic Rings

Let A be an analytic ring. We show: (1) A has finite Pythagoras number if and only if its real dimension is ≤ 2, and (2) if every positive semidefinite element of A is a sum of squares, then A is real and has real dimension 2.