Rings with No Nilpotent Elements and with the Maximum Condition on Annihilators

Abstract

Rings (all of which are assumed to be associative) with no non-zero nilpotent elements will be called reduced rings; R is a reduced ring if and only if x2=0 implies x=0, for all x∈R. In 2. we prove that the following conditions on an annihilator ideal I of a reduced ring are equivalent: I is a maximal annihilator, I is prime, I is a minimal prime, I is completely prime. A characterization of reduced rings with the maximum condition on annihilators is given in 3.

Coloring of lattices

“…In this paper we introduce and study the concept of coloring of a graph derived from a lattice with 0 (the smallest element) on the lines of Nimbhorkar, Wasadikar and DeMeyer [16] and characterize the chromatic number in the case of a graph derived from a distributive lattice. We also generalize some results from Cornish and Stewart [9] to distributive lattices with 0. The undefined terms and notations are from Grätzer [13] and Harary [15].…”

Reduced p.p.‐rings without identity

“…Then for all e ∈ E(R), L * (e) = Re is a right ideal of R. But e ∈ Re, so we have eR ⊆ Re. If f ∈ E(R), then e f = xe = e f e, where x ∈ R, hence e f = (e f ) 2 . This means that E(R) is a band.…”

“…(2)⇒(3)⇒(4) ⇒(5)⇒(2) is easy to see.For (1)⇒(2), we suppose that the ring R is reduced. Then ex = exe since (ex − exe) 2 = 0, for all e ∈ E(R) and x ∈ R. Similarly, xe = exe.…”

“…It can be observed that the class of p.p.-rings contains the classes of regular (von Neumann) rings, hereditary rings, Baer rings, and semi-hereditary rings as its proper subclasses. In the literature, p.p.-rings have already been studied by many authors (see [1,2,[5][6][7][8][9]11]). It is noteworthy that the definition of p.p.-rings has been extended to semigroups; in particular, Fountain [4] has introduced the concept of abundant semigroups which are both l.p.p.-and r.p.p.-semigroups.…”

“…On the other hand, Cornish and Stewart [2] called a ring R reduced if it contains no nonzero nilpotent elements. Obviously, the left annihilator ann (X) of X in a reduced ring R is always a two-sided ideal of R. Moreover, if R is a reduced ring, then e f = 0 if and only if f e = 0 for any nonzero idempotents e, f ∈ R. By using the concept of a reduced ring, Fraser and Nicholson [5] obtained an analogous result of Fountain's [4] that a ring R is a reduced p.p.-ring if and only if R is a (left; right) p.p.-ring in which every idempotent is central.…”