Classic algebra, by P. M. Cohn. Pp. 428. £27.50. 2000. ISBN 0 471 8773 1 X (Wiley). Classic algebra is a single-volume abridged version of Algebra by P. M. Cohn, published as a three volume set in 1973 and reprinted in 1981. It covers most of the algebra taught in mathematics courses at undergraduate level and some topics from graduate courses. There are eleven chapters in the book, each divided into sections. Many of these sections (particularly in the second half of the book) are self-contained, and in this way the book serves very well as a reference. Around a third of the book is devoted to linear algebra having comprehensive chapters on vector spaces, linear equations, determinants, quadratic forms and normal forms for matrices.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/22A03F0AD0BF67BCEBFF4EE36F09FD91/S0025557200170202a.pdf/div-class-title-span-class-italic-steps-in-commutative-algebra-span-2nd-edn-by-r-y-sharp-lms-student-texts-51-pp-355-17-95-2000-isbn-0-521-64623-5-cambridge-university-press-div.pdf

Steps in commutative algebra (2nd edn), by R. Y. Sharp. LMS Student Texts 51. Pp. 355. £17.95. 2000. ISBN 0 521 64623 5 (Cambridge University Press).

The purpose of the paper is to introduce and study weakly 2-prime ideals in commutative rings. Let A be a commutative ring with a nonzero identity. A proper ideal P of A is said to be a weakly 2-pr...

On weakly 2-prime ideals in commutative rings

This paper aims to introduce 2-absorbing φ -δ -primary ideals over commutative rings which unify the concepts of all generalizations of 2-absorbing and 2-absorbing primary ideals. Let A be a commutative ring with a nonzero identity and I(A) be the set of all ideals of A . Suppose that δ : I(A) → I(A) is an expansion function and φ : I(A) → I(A)∪{∅} is a reduction function. A proper ideal Q of A is said to be a 2-absorbing φ -δ -primary if whenever abc ∈ Q − φ(Q) , where a, b, c ∈ R, then either ab ∈ Q or ac ∈ δ(Q) or bc ∈ δ(Q). Various examples, properties, and characterizations of this new class of ideals are given.

2-Absorbing φ-δ -Primary Ideals

Contents: Functions of a Topological Space.- Ideals and Z-Filters.- Completely Regular Spaces.- Fixed Ideals. Compact Spaces.- Ordered Residue Class Rings.- The Stone-Czech Compactification.- Characterization of Maximal Ideals.- Realcompact Spaces.- Cardinals of Closed Sets in Beta-x.- Homomorphisms and Continuous Mappings.- Embedding in Products of Real Lines.- Discrete Spaces. Nonmeasurable Cardinals.- Hyper-Real Residue Class Fields.- Prime Ideals.- Uni- form Spaces.- Dimension.- Notes.- Bibliography.- List of Symbols.- Index.

Rings of continuous functions. By Leonard Gillman and Meyer Jerison. Pp. 300. 66s. 1960. (D. van Nostrand, London)

In this article, we introduce a generalization of the concept of graded $r$-ideals in graded commutative rings with nonzero unity. Let $G$ be a group, $R$ be a $G$-graded commutative ring with nonzero unity and $GI(R)$ be the set of all graded ideals of $R$. Suppose that $\phi: GI(R)\rightarrow GI(R)\bigcup\{\emptyset\}$ is a function. A proper graded ideal $P$ of $R$ is called a graded $\phi$-$r$-ideal of $R$ if whenever $x, y$ are homogeneous elements of $R$ such that $xy\in P-\phi(P)$ and $Ann(x) =\{0\}$, then $y\in P$. Several properties of graded $\phi$-$r$-ideals have been examined.

On Generalizations of Graded $r$-ideals

Let R be a commutative ring with nonzero identity and, S ⊆ R be a multiplicatively closed subset. An ideal P of R with P ∩ S = ∅ is called an S-prime ideal if there exists an (fixed) s ∈ S and when...

On S-Zariski topology

This paper introduce and study weakly 1-absorbing prime ideals in commutative rings. Let $$A\ $$
 be a commutative ring with a nonzero identity $$1\ne 0.$$
 A proper ideal $$P\ $$
 of $$A\ $$
 is said to be a weakly 1-absorbing prime ideal if for every nonunits $$x,y,z\in A\ $$
 with $$0\ne xyz\in P,\ $$
 then $$xy\in P$$
 or $$z\in P.\ $$
 In addition to give many properties and characterizations of weakly 1-absorbing prime ideals, we also determine rings in which every proper ideal is weakly 1-absorbing prime. Furthermore, we investigate weakly 1-absorbing prime ideals in C(X), which is the ring of continuous functions of a topological space $$X.\ $$

On Weakly 1-Absorbing Prime Ideals

Let G be a group with identity e and R be a G-graded commutative ring with unity 1. In this article, we introduce and study the concept of graded 1-absorbing prime ideals which is a generalization of graded prime ideals. A proper graded ideal P of R is called graded 1-absorbing prime if for all nonunit elements $$x,y,z\in h(R)$$
 such that $$xyz\in P$$
 , then either $$xy\in P$$
 or $$z\in P$$
 .

On graded 1-absorbing prime ideals

This paper introduce and study weakly 1-absorbing prime ideals in commutative rings. Let $A$ be a commutative ring with a nonzero identity $1\neq 0$. A proper ideal $P$ of $A$ is said to be a weakly 1-absorbing prime ideal if for each nonunits $x, y, z \in A$ with $0\neq xyz \in P$, then either $xy \in P$ or $z \in P$. In addition to give many properties and characterizations of weakly 1-absorbing prime ideals, we also determine rings in which every proper ideal is weakly 1-absorbing prime. Furthermore, we investigate weakly 1-absorbing prime ideals in $C(X)$, which is the ring of continuous functions of a topological space X.

Eda Yıldız

Papers

On S-Zariski topology

On Weakly 1-Absorbing Prime Ideals

On graded 1-absorbing prime ideals

On Weakly 1-Absorbing Prime Ideals

(2, j)-ideals in commutative rings