Mathematics

Let R be a commutative ring with nonzero identity and H be a nonempty proper subset of R such that R\H is a saturated multiplicatively closed subset of R. The generalized total graph of R is the (simple) graph GT H (R) with all elements of R as the vertices, and two distinct vertices x and y are adjacent if and only if x + y ∈ H. In this paper, we investigate the structure of GT H (R).

Let R be a commutative ring with identity 1 = 0. Various generalizations of prime ideals have been studied. For example, a proper ideal I of R isweakly prime if a, b ∈ R with 0 = ab ∈ I, then either a ∈ I or b ∈ I. Also a proper ideal I of R is said to be 2-absorbing if whenever a, b, c ∈ R and abc ∈ I, then either ab ∈ I or ac ∈ I or bc ∈ I. In this paper, we introduce the concept of a weakly 2-absorbing ideal. A proper ideal I of R is called a weakly 2-absorbing ideal of R if whenever a, b, c ∈ R and 0 = abc ∈ I, then either ab ∈ I or ac ∈ I or bc ∈ I. For example, every proper ideal of a quasi-local ring (R, M ) with M 3 = {0} is a weakly 2-absorbing ideal of R. We show that a weakly 2-absorbing ideal I of R with I 3 = 0 is a 2-absorbing ideal of R. We show that every proper ideal of a commutative ring R is a weakly 2-absorbing ideal if and only if either R is a quasi-local ring with maximal ideal M such that M 3 = {0} or R is ringisomorphic to R 1 × F where R 1 is a quasi-local ring with maximal ideal M such that M 2 = {0} and F is a field or R is ring-isomorphic to F 1 × F 2 × F 3 for some fields F 1 , F 2 , F 3 .

Synonyms 6 Addition and subtraction of matrices; Augmented 7 matrix; Consistent and inconsistent system; 8 Cramer rule; Determinant of a matrix; Echelon 9 form; Elementary matrix; Invertible (nonsin-10 gular) matrices; Multiplication of matrices; 11 Symmetric and skew-symmetric matrices; 12 Transpose of a matrix 13 Glossary 14 Matrix n m A matrix is a block consisting of 15 n row and m column. An entry in a matrix B 16 located in the i th row and j th column of B is 17 denoted by b ij 18 Consistent and Inconsistent System of Linear 19 Equations A system of linear equations is 20 said to be consistent if it has a solution, and 21 it is called inconsistent if it has no solutions 22 Augmented Matrix An augmented matrix of a 23 system of linear equations written in matrix-24 form CX D B is a matrix of the form OEC jB, 25 where C is the coefficient matrix of the system 26 and B is the constant column of the system 27

Let R be a commutative ring with Nil(R) its ideal of nilpotent elements, Z(R) its set of zero-divisors, and Reg(R) its set of regular elements. In this paper, we introduce and investigate the total graph of R, denoted by T (Γ (R)). It is the (undirected) graph with all elements of R as vertices, and for distinct x, y ∈ R, the vertices x and y are adjacent if and only if x + y ∈ Z(R). We also study the three (induced) subgraphs Nil(Γ (R)), Z(Γ (R)), and Reg(Γ (R)) of T (Γ (R)), with vertices Nil(R), Z(R), and Reg(R), respectively.

All rings considered are commutative with identity. We study the preservation of certain properties in the passage from a ring R to the ultrapower R relative to a free ultrafilter on the set of all positive integers. Our main result is that if R is a locally pseudo-valuation domain (LPVD) of finite character (for instance, a semi-quasilocal LPVD), then R is also an LPVD. In the same vein, it is shown that the classes of pseudo-valuation domains and pseudo-valuation rings are each stable under the passage from R to R . An example is given of a divided domain R such that the domain R is not divided. A divisibility condition is found which characterizes the divided (respectively, quasilocal) rings R such that R is a divided (respectively, treed) ring.

Let R be an integral domain with quotient field K and integral closure R . Anderson and Zafrullah called R an "almost valuation domain" if for every nonzero x ∈ K, there is a positive integer n such that either x n ∈ R or x −n ∈ R. In this article, we introduce a new closely related class of integral domains. We define a prime ideal P of R to be a "pseudo-strongly prime ideal" if, whenever x y ∈ K and xyP ⊆ P, then there is a positive integer m ≥ 1 such that either x m ∈ R or y m P ⊆ P.

A nonzero ideal I of an intergral domain R is said to be an m-canonical ideal of R if ðI : ðI : JÞÞ ¼ J for every nonzero ideal J of R. In this paper, we show that if a quasi-local integral domain ðR; MÞ admits a proper m-canonical ideal I of R, then the following statements are equivalent:

Let R be a commutative ring with 1 6 D 0 and Nil.R/ be its set of nilpotent elements. Recall that a prime ideal of R is called a divided prime if P .x/ for every x 2 R n P ; thus a divided prime ideal is comparable to every ideal of R. In many articles, the author investigated the class of rings H D ¹R j R is a commutative ring and Nil.R/ is a divided prime ideal of Rº (Observe that if R is an integral domain, then R 2 H .) If R 2 H , then R is called a -ring. Recently, David Anderson and the author generalized the concept of PrR ufer domains, Bezout domains, Dedekind domains, and Krull domains to the context of rings that are in the class H . Also, Lucas and the author generalized the concept of Mori domains to the context of rings that are in the class H . In this paper, we state many of the main results on -rings.