Abstract: Let be a commutative ring and be a unitary module. We define a semiprime submodule of a module and consider various properties of it. Also we define semi-radical of a submodule of a module and give a number of its properties. We define modules which satisfy the semi-radical formula and present the existence of such a module.  

We introduce an addition law for the usual quantummatrices A(R) by means of a coaddition ∆t = t⊗ 1 + 1⊗ t. It supplements the usual comultiplication ∆t = t⊗ t and together they obey a codistributivity condition. The coaddition does not form a usual Hopf algebra but a braided one. The same remarks apply for rectangular m × n quantum matrices. As an application, we construct leftinvariant vector ...

The self-dual concept of a quantum quasigroup offers a far-reaching unification of Hopf algebras (along with their non-associative extensions, e.g., [1–4]) and quasigroups [5]. Consider a bimagma (A,∇, ∆), an object of a strict symmetric monoidal category V with V-morphisms giving a magma structure or multiplication ∇ : A⊗ A→ A, and a comagma structure or comultiplication ∆ : A→ A⊗ A, such that...

We introduced a non-symmetric tensor product of any two states or any two representations of Cuntz-Krieger algebras associated with a certain non-cocommutative comultiplication in previous our work. In this paper, we show that a certain set of KMS states is closed with respect to the tensor product. From this, we obtain formulae of tensor product of type III factor representations of Cuntz-Krie...

An R-module M is called epi-retractable if every submodule of MR is a homomorphic image of M. It is shown that if R is a right perfect ring, then every projective slightly compressible module MR is epi-retractable. If R is a Noetherian ring, then every epi-retractable right R-module has direct sum of uniform submodules. If endomorphism ring of a module MR is von-Neumann regular, then M is semi-...

The concepts of free modules, projective modules, injective modules and the likeform an important area in module theory. The notion of free fuzzy modules was introducedby Muganda as an extension of free modules in the fuzzy context. Zahedi and Ameriintroduced the concept of projective and injective L-modules. In this paper we give analternate definition for projective L-modules. We prove that e...

The purpose of this paper is to give definitions of some new Hopf algebras analogous to the mod-p Steenrod algebras. For example, we produce a Hopf algebra over the integers whose reduction mod-p is the mod-p Steenrod algebra. Just for fun, we “quantize” these algebras too! Further study of the examples of this paper will be done in part II. 1 Deformations of Hopf Algebras Suppose that k is a c...

we show that every semi-artinian module which is contained in a direct sum of finitely presented modules in $si[m]$, is weakly co-semisimple if and only if it is regular in $si[m]$. as a consequence, we observe that every semi-artinian ring is regular in the sense of von neumann if and only if its simple modules are $fp$-injective.

let r be a ring,  be an endomorphism of r and mr be a -rigid module. amodule mr is called quasi-baer if the right annihilator of a principal submodule of r isgenerated by an idempotent. it is shown that an r-module mr is a quasi-baer module if andonly if m[[x]] is a quasi-baer module over the skew power series ring r[[x; ]].