let r be a commutative ring with identity. let n and k be two submodules of a multiplication r-module m. thenn=im and k=jm for some ideals i and j of r. the product of n and k denoted by nk is defined by nk=ijm. inthis paper we characterize some particular cases of multiplication modules by using the product of submodules.

let  be a banach algebra and  a derivation. in this paper, it is proved, under certain conditions, that , where  is the jacobson radical of . moreover, we prove that if  is unital and  is a continuous derivation, then , where  denotes the set of all primitive ideals such that  is commutative,  denotes the set of all maximal (modular) ideals such that  is commutative, and  is the set of all non-...

we state several conditions under which comultiplication and weak comultiplication modulesare cyclic and study strong comultiplication modules and comultiplication rings. in particular,we will show that every faithful weak comultiplication module having a maximal submoduleover a reduced ring with a finite indecomposable decomposition is cyclic. also we show that if m is an strong comultiplicati...

In this paper, by using elementary tools of commutative algebra,we prove the persistence property for two especial classes of rings. In fact, thispaper has two main sections. In the first main section, we let R be a Dedekindring and I be a proper ideal of R. We prove that if I1, . . . , In are non-zeroproper ideals of R, then Ass1(Ik11 . . . Iknn ) = Ass1(Ik11 ) [ · · · [ Ass1(Iknn )for all k1,...

The concepts of L-fuzzy ideal generated by a L-fuzzy subset, L-fuzzy prime and completely prime ideal where L is a complete lattice are considered and some results are proved

Let I be a monomial squarefree ideal of a polynomial ring S over a field K such that the sum of every three different ideals of its minimal prime ideals is the maximal ideal of S, or more generally a constant ideal. We associate to I a graph on [s], s = |MinS/I|, on which we may read the depth of I. In particular, depthS I does not depend on char K. Also we show that I satisfies Stanley’s Conje...

Under Greenberg’s conjecture, we give an efficient method to compute the p-part of the ideal class group of certain real abelian fields by using cyclotomic units, Gauss sums and prime numbers. As numerical examples, we compute the p-part of the ideal class group of the maximal real subfield of Q( √ −f, ζpn+1) in the range 1 < f < 200 and 5 ≤ p < 100000. In order to explain our method, we show a...

Suppose M is a maximal ideal of a commutative integral domain R and that some power Mn of M is finitely generated. We show that M is finitely generated in each of the following cases: (i) M is of height one, (ii) R is integrally closed and htM = 2, (iii) R = K[X; S̃] is a monoid domain over a field K, where S̃ = S ∪ {0} is a cancellative torsion-free monoid such that ⋂∞ m=1 mS = ∅, and M is the m...

2 Integral Domains 12 2.1 Factorization in Integral Domains . . . . . . . . . . . . . . . . 12 2.2 Euclidean Domains . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Principal Ideal Domains . . . . . . . . . . . . . . . . . . . . . 16 2.4 Fermat’s Two Squares Theorem . . . . . . . . . . . . . . . . . 17 2.5 Maximal Ideals and Prime Ideals . . . . . . . . . . . . . . . . 20 2.6 Unique Fact...

Let O be a maximal order in a definite quaternion algebra over Q of prime discriminant p, and ` a small prime. We describe a probabilistic algorithm, which for a given left O-ideal, computes a representative in its left ideal class of `-power norm. In practice the algorithm is efficient, and subject to heuristics on expected distributions of primes, runs in expected polynomial time. This breaks...