We introduce a graph structure \(\Gamma^{\ast}_2(R)\) for commutative rings with unity. study some of the properties \(\Gamma^{\ast}_2(R)\). Also we parameters and find which is split.

ideals of these kinds; congruence class semirings. Abstract: Since they are closer related to the ring-theoretical concept of ideals than arbitrary semiring ideals, two-sided k-ideals and h-ideals occur in several statements on semirings. In this paper we investigate those ideals and also left-sided ones in arbitrary semirings S with commutative addition. For instance, we prove (for the rst tim...

The method of Gröbner bases in commutative polynomial rings introduced by Buchberger (e.g. [1]) is a well-known and very important tool in polynomial ideal theory, for example in solving the ideal membership problem. XIDEAL extends the method to exterior algebras using algorithms from [2] and [3]. There are two main departures from the commutative polynomial case. First, owing to the non-commut...

An approach is proposed for inferring implicative logical rules from examples. The concept of a good diagnostic test for a given set of positive examples lies in the basis of this approach. The process of inferring good diagnostic tests is considered as a process of inductive common sense reasoning. The incremental approach to learning algorithms is implemented in an algorithm DIAGaRa for infer...

The aim of this short note is to answer a question by Guoliang Yu of whether the group EL3(Z〈x, y〉), where Z〈x, y〉 is the free (non-commutative) ring, has any faithful finite dimensional linear representations over a field. Recall that for every (associative unitary) ring R the group ELn(R) is the subgroup of GLn(R) generated by all n × n-elementary matrices xij(r) = Id + reij (r ∈ R, 1 ≤ i 6= ...

INTRODUCTION Recently, the use of Groebner bases and Buchberger algorithm [BUC1,2,4] has been generalised from the case of commutative polynomials to finitely generated algebras R over a field k, R = k, s.t. for each i < j, for some cij ∈ k, for some commutative polynomial pij ∈ k[X1,...,Xn], one has xj xi cij xi xj = pij(x1,...,xn). The first results in this direction were due to Ga...