Ralph P. Tucci

Zero Divisor Graphs of Upper Triangular Matrix Rings

Let R be a commutative ring with identity 1 ≠ 0 and T be the ring of all n × n upper triangular matrices over R. In this paper, we describe the zero divisor graph of T. Some basic graph theory properties of are given, including determination of the girth and diameter. The structure of is discussed, and bounds for the number of edges are given. In the case that R is a finite integral domain and n = 2, the structure of is fully described and an explicit formula for the number of edges is given.

The Circle Semigroup of a Ring

Let R be a ring and define x ○ y = x + y - xy, which yields a monoid (R, ○), called the circle semigroup of R. This paper investigates the relationship between the ring and its circle semigroup. Of particular interest are the cases where the semigroup is simple, 0-simple, cancellative, 0-cancellative, regular, inverse, or the union of groups, or where the ring is simple, regular, or a domain. The idempotents in R coincide with the idempotents in (R, ○) and play an important role in the theory developed.

Zero Divisor Graphs of Finite Direct Products of Finite Rings

We determine the number of edges of the finite direct product of finite rings. We apply this result to finite rings without idempotents, in particular direct products of ℤ m .

Right Weakly Regular Rings: A Survey

A ring is right weakly regular (r.w.r.) if every right ideal of the ring is idempotent. Such rings are also called fully right idempotent. This paper gives a survey of the theory of r.w.r. rings and some closely allied topics, from its origins in the early 1950’s up to the present state-of-the-art. The paper contains sections on: equivalent conditions, examples and constructions, and related conditions.

Adjoint regular rings

Let R be a ring. The circle operation is the operation a ∘ b = a + b − a b , for all a , b ∈ R . This operation gives rise to a semigroup called the adjoint semigroup or circle semigroup of R . We investigate rings in which the adjoint semigroup is regular. Examples are given which illustrate and delimit the theory developed.

Expressing quantification in relational calculus by participation constraints

Abstract In this paper, we show that in relational calculus some expressions with quantification are equivalent to expressions with participation constraints. This result leads to a simple way to express many common queries that are considered difficult.

A note on the decomposition of infinite automata

We derive a decomposition theorem for infinite automata similar to the Krohn-Rhodes decomposition theorem.

The Krull radical, k-primitive rings, and critical rings

We generalize results on the Krull radical, k-primitive rings, and critical rings from rings with identity to rings which do not necessarily contain identity..

MAXIMAL IDEALS IN RINGS

This paper gives necessary and sufficient conditions which guarantee that a ring have maximal left, right, or two-sided ideals. The relations of rings without maximal ideals to the Jacobson and Brown-McCoy radical are discussed. Examples of rings without maximal ideals are given to illustrate the theory. Connections of existence of maximal ideals and the Axiom of Choice in Zermelo-Fraenkel set theory are noted.

Inverse semigroups all of whose proper homomorphic images are groups

The bicyclic semigroup is defined as C {p, q \ pq = 1). It is well known [2, Corollary 3.2], that every proper (non-isomorphic) homomorphic image of the bicyclic semigroup C is a group. (In fact, every proper homomorphic image of C is a cyclic group; however, we shall not use the cyclic property.) We shall refer to inverse semigroups all of whose proper homomorphic images are groups as h-groups; to eliminate certain trivial cases, we shall require that an /i-group S does have homomorphic images other than itself and the one-element semigroup, that S is not a group, and that 5 has more than two elements. In this paper we characterise ft-groups in general. We also show that the bicyclic semigroup is the only /i-group in certain cases.