A ring R is called an E-ring if the canonical homomorphism from R to the endomorphism ring End (RZ) of the additive group RZ, taking any r ∈ R to the endomorphism left multiplication by r turns out to be an isomorphism of rings. In this case RZ is called an E-group. Obvious examples of E-rings are subrings of Q. However there is a proper class of examples constructed recently, see [8]. E-rings ...

An significant milestone study in coding theory recognized to be the paper written by Hammons at al. [1]. Fields are useful area for constructing codes but after the study [1] finite ring have received a great deal of attention. Most of the studies are concentrated on the case with codes over finite chain rings. However, optimal codes over nonchain rings exist (e.g see [2].) In [3], et al. stud...

Let J be a graded ideal in a not necessarily commutative graded k-algebra A = A0⊕A1⊕· · · in which dimk Ai <∞ for all i. We show that the map A→ A/J induces a closed immersion i : Projnc A/J → Projnc A between the non-commutative projective spaces with homogeneous coordinate rings A and A/J . We also examine two other kinds of maps between non-commutative spaces. First, a homomorphism φ : A → B...

Linear network coding over finite fields is a wellstudied problem. We consider the more general setting of linear coding for directed acyclic networks with finite commutative ring alphabets. Our results imply that for scalar linear network coding over commutative rings, fields can always be used when the alphabet size is flexible, but other rings may be needed when the alphabet size is fixed. W...

Let G be a group and R G-graded ring. In this paper, we present examine the concept of graded weakly 2-absorbing ideals as in generality prime ring which is not commutative, demonstrates that symmetry obtained lot outcomes commutative rings remain are commutative.