JOURNAL OF ALGEBRAIC SYSTEMS
Year:2015 | Volume:3 | Issue:1 |Papers Count:8

generalization of the notion of depth of an ideal on a module is introduced by applying the concept of local cohomology modules with respect to a pair of ideals. The concept of (I; J)-Cohen{Macaulay modules is also introduced as a generalization of the concept of Cohen{Macaulay modules. This kind of modules is di erent from the Cohen{Macaulay modules, as shown in an example. Also an Artinian result is given for such modules.

In this paper, a generalization of the integral depen-dence of rings on modules is given. The stability of the integral closure with respect to various module theoretic constructions is also studied. Moreover, the notion of integral extension of a mod-ule is introduced, and the Lying over, Going up and Going down theorems for modules are proved.

The generalized principal Ideal theorem is one of the cornerstones of the dimension theory for the Noetherian rings. For an R-module M, certain submodules of M that play a role analo- gous to that of prime ideals in the ring R are identied. Using this denition, we extend the this theorem to modules.

The rank-k numerical range has a close connection to the construction of quantum error correction code for a noisy quantum channel. For a noisy quantum channel, a quantum error correcting code of dimension k exists, if and only if the associated joint rank-k numerical range is non-empty. In this paper, the no-tion of joint rank-k numerical range is generalized, and some state-ments of [2011, Generalized numerical ranges and quantum error correction, J. Operator Theory, 66: 2, 335-351. ] are extended.

In this article, we give several generalizations of the concept of annihilating an ideal graph over a commutative ring with identity to modules. We observe that, over a commutative ring, R, AG (RM) is connected, and diamAG (RM) ≤ 3. More-over, if AG (RM) contains a cycle, then grAG (RM) ≤ 4. Also for an R-module M with A (M) ̸ = S(M) \ {0}, A (M) = ∅ , if and only if M is a uniform module, and ann(M) is a prime ideal of R.

In this paper, we continue our study on HvMV-algebras. The quotient structure of an HvMV-algebra by a suitable type of congruence is studied, and some properties and related results are given. Some homomorphism theorems are given, as well. Also the fundamental HvMV-algebra, and the direct product of a family of HvMV-algebras are investigated, and some related results are obtained.

In this paper, the fuzzy subnexuses over a nexus N are de ned and the notions of prime fuzzy subnexuses and fractions induced by them are studied. Finally, it is shown that if S is a meet closed subset of the set Fsub(N), of fuzzy subnexuses of a nexus N, and h = ∧ S ∈ S, then the fractions S􀀀 1N and {h}􀀀 1N are isomorphic as meet-semilattices.

Suppose G is a  nite group, A and B are conjugacy classes of G, and  (AB) denotes the number of conjugacy classes contained in AB. The set of all  (AB), such that A; B run over conjugacy classes of G, is denoted by  (G). The aim of this paper is to compute  (G), for G ∈ {D2n; T4n; U6n; V8n; SD8n} or G is a decomposable group of order 2pq, a group of order 4p or p3, where p and q are primes.