Attached primes over noncommutative rings

Prime fuzzy ideals over noncommutative rings

In this paper we introduce prime fuzzy ideals over a noncommutative ring. This notion of primeness is equivalent to level cuts being crisp prime ideals. It also generalizes the one provided by Kumbhojkar and Bapat in [16], which lacks this equivalence in a noncommutative setting. Semiprime fuzzy ideals over a noncommutative ring are also defined and characterized as intersection of primes. This...

Linear Equations over Noncommutative Graded Rings

We call a graded connected algebra R effectively coherent, if for every linear equation over R with homogeneous coefficients of degrees at most d, the degrees of generators of its module of solutions are bounded by some function D(d). For commutative polynomial rings, this property has been established by Hermann in 1926. We establish the same property for several classes of noncommutative alge...

Computing minimal associated primes in polynomial rings over the integers

An algorithm is presented to compute the minimal associated primes of an ideal in a polynomial ring over the integers. It differs from the known algorithms insofar as it avoids having to compute Gröbner bases over the integers until the very end, thereby eliminating one of the bottlenecks of those algorithms. © 2011 Elsevier Ltd. All rights reserved.

Lie algebras and Lie groups over noncommutative rings

The aim of this paper is to introduce and study Lie algebras and Lie groups over noncommutative rings. For any Lie algebra g sitting inside an associative algebra A and any associative algebra F we introduce and study the algebra (g,A)(F), which is the Lie subalgebra of F ⊗ A generated by F ⊗ g. In many examples A is the universal enveloping algebra of g. Our description of the algebra (g,A)(F)...

Three Examples of Gröbner Reduction over Noncommutative Rings