This text, drawn from the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-term course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semi-simple rings, Jacobson's theory of the radical representation theory of groups and algebras, prime and semi-prime rings, primitive and semi-primitive rings, division rings, ordered rings, local and semi-local rings, and perfect and semi-perfect rings. By aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation, the author has produced a text which is suitable not only for use in a graduate course, but also for self-study by other interested graduate students. Numerous exercises are also included. This graduate textbook on rings, fields and algebras is intended for graduate students in mathematics.

A first course in noncommutative rings, by T. Y. Lam. Pp. 385. £37 (pb), £62.50 (hb). 2001. ISBN 0 387 95325 6 (pb), 0 387 95183 0 (hb) (Springer-Verlag).

https://sciold.ui.ac.ir/~a.abdollahi/PDF/comm-graph.pdf

Non-commuting graph of a group ✩

A signed graph is a graph whose edges are labeled by signs. This is a bibliography of signed graphs and related mathematics. Several kinds of labelled graph have been called "signed" yet are mathematically very different. I distinguish four types: Group-signed graphs: the edge labels are elements of a 2-element group and are multiplied around a polygon (or along any walk). Among the natural generalizations are larger groups and vertex signs. Sign-colored graphs, in which the edges are labelled from a two-element set that is acted upon by the sign group: - interchanges labels, + leaves them unchanged. This is the kind of "signed graph" found in knot theory. The natural generalization is to more colors and more general groups —  or no group. Weighted graphs, in which the edge labels are the elements +1 and -1 of the integers or another additive domain. Weights behave like numbers, not signs; thus I regard work on weighted graphs as outside the scope of the bibliography — ex cept (to some extent) when the author calls the weights "signs". Labelled graphs where the labels have no structure or properties but are called "signs" for any or no reason.

/pdf/a-mathematical-bibliography-of-signed-and-gain-graphs-and-3a49f10hs2.pdf

A Mathematical Bibliography of Signed and Gain Graphs and Allied Areas

This article surveys the recent and active area of zero-divisor graphs of commutative rings. Notable algebraic and graphical results are given, followed by a historical overview and an extensive bibliography.

Zero-divisor graphs in commutative rings

Let R be a commutative ring, with &#x1d538;(R) its set of ideals with nonzero annihilator. In this paper and its sequel, we introduce and investigate the annihilating-ideal graph of R, denoted by &#x1d538;&#x1d53e;(R). It is the (undirected) graph with vertices &#x1d538;(R)* ≔ &#x1d538;(R)\{(0)}, and two distinct vertices I and J are adjacent if and only if IJ = (0). First, we study some finiteness conditions of &#x1d538;&#x1d53e;(R). For instance, it is shown that if R is not a domain, then &#x1d538;&#x1d53e;(R) has ascending chain condition (respectively, descending chain condition) on vertices if and only if R is Noetherian (respectively, Artinian). Moreover, the set of vertices of &#x1d538;&#x1d53e;(R) and the set of nonzero proper ideals of R have the same cardinality when R is either an Artinian or a decomposable ring. This yields for a ring R, &#x1d538;&#x1d53e;(R) has n vertices (n ≥ 1) if and only if R has only n nonzero proper ideals. Next, we study the connectivity of &#x1d538;&#x1d53e;(R). It is shown that &#x1d538;&#x1d53e;(R) is a connected graph and diam(&#x1d538;&#x1d53e;)(R) ≤ 3 and if &#x1d538;&#x1d53e;(R) contains a cycle, then gr(&#x1d538;&#x1d53e;(R)) ≤ 4. Also, rings R for which the graph &#x1d538;&#x1d53e;(R) is complete or star, are characterized, as well as rings R for which every vertex of &#x1d538;&#x1d53e;(R) is a prime (or maximal) ideal. In Part II we shall study the diameter and coloring of annihilating-ideal graphs.

A. Mohammadian

Papers

On the zero-divisor graph of a commutative ring

Zero-divisor graphs of non-commutative rings

On the diameters of commuting graphs

On zero-divisor graphs of finite rings

On commuting graphs of semisimple rings