Goldie’s Theorem (1960), which is one of the most important results in Ring Theory, is a criterion for a ring to have a semisimple left quotient ring. The aim of the paper is to give four new criteria (using a completely different approach and new ideas). The first one is based on the recent fact that for an arbitrary ring R the set M of maximal left denominator sets of R is a non-empty set [2]...

Let S = k[x1, . . . , xn] be a polynomial ring over a field k and I a monomial ideal of S. It is well known that the Poincaré series of k over S/I is rational. We describe the coefficients of the denominator of the series and study the multigraded homotopy Lie algebra of S/I.

It is shown that a commutative Bézout ring R with compact minimal prime spectrum is an elementary divisor ring if and only if so is R/L for each minimal prime ideal L. This result is obtained by using the quotient space pSpec R of the prime spectrum of the ring R modulo the equivalence generated by the inclusion. When every prime ideal contains only one minimal prime, for instance if R is arith...

We construct reduced Groebner bases for a certain class of ideals in commutative polynomial rings. A subclass of these ideals corresponds to the generalized Reed-Muller codes when considered in the quotient ring of the polynomial ring. AMS Subject Classification: 13P10, 94B30

The aim of this paper is to study the concept of Intuitionistic L-fuzzy subrings and Intuitionistic L-fuzzy ideals of a ring R. In this direction definitions and properties relating to Intuitionistic L-fuzzy subrings of R and Intuitionistic L-fuzzy ideals of R are introduced and discussed. We introduce a special type of Quotient ring.

We construct a quotient ring of the ring of diagonal coinvariants of the complex reflection group W = G(m, 1, n) and determine its graded character. This generalises a result of Gordon for Coxeter groups. The proof uses a study of category O for the rational Cherednik algebra of W , including a shift isomorphism which is proved in Appendix 1.

We use moduli spaces for covers of the Riemann sphere to solve regular embedding problems, with prescribed extendability of orderings, over PRC fields. As a corollary we show that the elementary theory of Qtr is decidable. Since the ring of integers of Qtr is undecidable, this gives a natural undecidable ring whose quotient field is decidable.

Gaussian integer is one of basic algebraic integers. In this article we formalize some definitions about Gaussian integers [27]. We also formalize ring (called Gaussian integer ring), Z-module and Z-algebra generated by Gaussian integer mentioned above. Moreover, we formalize some definitions about Gaussian rational numbers and Gaussian rational number field. Then we prove that the Gaussian rat...

We study an RG-module A , where R is a ring, A/CA(G) is infinite, CG(A) = 1, G is a group. Let Lnf(G) be the system of all subgroups H ≤ G such that the quotient modules A/CA(H) are infinite. We investigate an RG-module A such that Lnf(G) satisfies either the weak minimal condition or the weak maximal condition as an ordered set. It is proved that if G is a locally finite group then either G is...

Gaussian integer is one of basic algebraic integers. In this article we formalize some definitions about Gaussian integers [27]. We also formalize ring (called Gaussian integer ring), Z-module and Z-algebra generated by Gaussian integer mentioned above. Moreover, we formalize some definitions about Gaussian rational numbers and Gaussian rational number field. Then we prove that the Gaussian rat...