Valuation rings with zero divisors

Prüfer Conditions in Rings with Zero- Divisors

In his article: “Untersuchungen über die Teilbarkeitseigenschaften in Körpern” J. Reine Angew. Math. 168, 1 36, 1932 [21], Heinz Prüfer introduced a new class of integral domains, namely those domains R in which all finitely generated ideals are invertible. He also proved that to verify this condition, it suffices to check that it holds for all two-generated ideals of R. This was the modest beg...

Characterizations of Krull Rings with Zero Divisors

We show that a ring is a Krull ring if and only if every nonzero regular prime ideal contains a t-invertible prime ideal if and only if every proper regular principal ideal is quasi-equal to a product of prime ideals.

Zero Divisors and Commutativity of Rings

Divisors of Zero in Matric Rings

Smarandache-Zero Divisors in Group Rings

The study of zero-divisors in group rings had become interesting problem since 1940 with the famous zero-divisor conjecture proposed by G.Higman [2]. Since then several researchers [1, 2, 3] have given partial solutions to this conjecture. Till date the problem remains unsolved. Now we introduce the notions of Smarandache zero divisors (S-zero divisors) and Smarandache week zero divisors (S-wea...