Regular rings (Q1106888)

Although there are numerous conditions in the literature which characterize regular rings, these usually involve some sort of (quasi)- ideal-theoretic intersection versus product conditions. In this paper, the set of interesting subsets of a commutative ring R with identity is the set of saturated subsets F (i.e., xy\(\in F\) iff \(x\in F, y\in F)\) of R denoted by S(R), whose properties are explored. Thus, R is regular iff R is semiprime and [a) (the smallest saturated set containing a) has a complement in the lattice S(R) theorem 1); R is Noetherian regular iff S(R) is also a Boolean algebra (theorem 2). Further variations on this theme are also developed in this useful paper.