Algebras With No Commutative Relations And Zero Divisor Graphs

Using the graph form to describe algebraic structures, which can help us not only enable the abstract knowledge explicit, but also use the structure of graphs to characterize the property of algebras. The quiver theory and zero divisor graphs are one of representations of these kinds. Employing this methods on this thesis, we study the generalization of monomial algebras, the proof of Nakayama conjecture and the graph structure of von Neumann regular rings, which consisting of three chapters.Chaper 1 introduces the background, methods and main results obtained in this thesis.We mainly discuss the generalization of monomial algebras in chapter 2, and define a kind of new algebras-algebras with no commutative relations, whose basic properties will be studied and the Nakayama conjecture, Auslander-Reiten conjec-ture will be proved on this kind of algebras, and hence generalized the results about these conjectures.As an open problem, the generalization of monomial algebras is an important problem (see [14]), since not only it has been the starting point of many questions because of its well understand structures, but also many homological conjectures are known to be true for these algebras (see [28] [29]).According to the methods and means in quiver theory and path algebras, we define a class new algebras which contains no commutative relations by in-troduce some notions and methods on artin algebras which look like the one on path algebras in (?)2.2, and call it the algebra with no commutative relations (see Definition2.2.9).In (?)2.3, we get an important conclusion, that is, if K is a field, and∧(?)(?)Kг/I is monomial, then A is an algebra with no commutative relations (see Theo-rem2.3.5), which shows that the monomial algebra is a special case of the algebra with no commutative relations, moreover, we list several important examples which help us show that the converse is not true. Thus we prove that this generalization of monomial algebras is non-trivial. In (?)2.4 and (?)2.5, we prove that the Nakayama conjecture and Auslander-Reiten conjecture are true for this class algebras which is a generalization of mono-mial algebras. We deal the elements of right artinian algebra A with the homo-morphisms between project modules, and combine with quiver theory, study the properties of right artinian algebras of dominant dimension larger than or qual to 1 (or 2). Consequently, we obtain the most important conclusion of this thesis,that is, right artinian algebra A with dominant dimension larger than or equal to 2 is the Nakayama algebra if and only if it is the algebra with no commutative relations (see Theorem2.5.9). Thus, it is not hard to check that the Nakayama conjecture and Auslander-Reiten conjecture are true for this kind of algebras.Chapter 3 mainly study the graph structure on von Neumann regular rings.The studying of zero-divisor graphs, on one hand, it affords many new meth-ods to study the structure and property of algebraic systems, on the other hand, it gives us more objects and applied platform for studying the graph theory.Recently, all studying of directed zero-divisor graphs is almost concerned with finite rings (see), for the directed zero-divisor graphг(R) of von Neumann regular ring R, (?)3.2 characterizes its connectedness and vertices, we prove that for a von Neumann regular ring R, its directed zero-divisor graphг(R) is connected if and only if R is direct finite, moreover, if R has no identity, thenг(R) is connected if and only if R has no proper one-side identity. Let Sour(R), Sink(R) denote the set of sources and sinks of T(R), respectively. Then we have:(1) Sour(R)={a∈R|a is right but not left invertible};(2) Sink(R)={a∈R|a is left but not right invertiblc}.Employing the methods of zero-divisor graphs on (?)3.3, we introduce a kind of new graph structure FN(R) on an arbitrary ring R, this kind of graph structure contains more information about nilpotent elements than zero-divisor graphs. For a von Neumann regular ring R, we characterize the connectedness, diameter and girth ofгN(R) (see Theorem3.3.6), also we determine all non-reduced von Neumann regular rings for whichгN(R) is a star graph (see Theorem3.3.7). For a non-reduced finite commutative ring R, we get the edge coloring formula, that is, x'(гN(R))= △(гN(R)),unless R=N(R)and |R| is an even number(see Theorem3.3.14). Simultaneously,if S is a commutative ring but not a field,then with two exceptions,гN(R)(?)гN(S)(?)R(?)S(see Theorem3.3.16).