Non-Commutative Ring Theory Research Papers (original) (raw)

In this paper we define an E1-structure, i.e. a coherently homotopy associative and commutative product on chain complexes defining (integral and mod-l) motivic cohomology as well as mod -letale cohomology. We also discuss several... more

In this paper we define an E1-structure, i.e. a coherently homotopy associative and commutative product on chain complexes defining (integral and mod-l) motivic cohomology as well as mod -letale cohomology. We also discuss several applications.

JETIR2108239 Journal of Emerging Technologies and Innovative Research (JETIR) www.jetir.org b871 Exploring the beginning of Algebraic KTheory Alok Prasad Rout, N. Jagannadham Research Scholar, Department of Mathematics, GIET University,... more

JETIR2108239 Journal of Emerging Technologies and Innovative Research (JETIR) www.jetir.org b871 Exploring the beginning of Algebraic KTheory Alok Prasad Rout, N. Jagannadham Research Scholar, Department of Mathematics, GIET University, Gunupur 2 Assistant Professor, Department of Mathematics, GIET University, Gunupur Abstract According to W. Keith, K-theory is that part of linear algebra that studies additive or abelian properties (e.g. the determinant). Because linear algebra, and its extensions to linear analysis, is ubiquitous in mathematics, K-theory has turned out to be useful and relevant in most branches of mathematics. Let R be a ring. One defines K0(R) as the free abelian group whose basis are the finitely generated projective R-modules with the added relation P ⊕ Q = P + Q. The purpose of this thesis is to study simple settings of the K-theory for rings and to provide a sequence of examples of rings where the associated K-groups K0(R) get progressively more complicated. W...

summary:Let RRR be a ring with an identity (not necessarily commutative) and let MMM be a left RRR-module. This paper deals with multiplication and comultiplication left RRR-modules MMM having right operatornameEndR(M)\operatorname{End}_R(M)operatornameEndR(M)-module... more

summary:Let RRR be a ring with an identity (not necessarily commutative) and let MMM be a left RRR-module. This paper deals with multiplication and comultiplication left RRR-modules MMM having right operatornameEndR(M)\operatorname{End}_R(M)operatornameEndR(M)-module structures

‎In this paper‎, ‎we introduce the concepts of nnn-absorbing and strongly nnn-absorbing second submodules as a dual notion of nnn-absorbing submodules of modules over a commutative ring and obtain some related results‎. ‎In particular‎,... more

‎In this paper‎, ‎we introduce the concepts of nnn-absorbing and strongly nnn-absorbing second submodules as a dual notion of nnn-absorbing submodules of modules over a commutative ring and obtain some related results‎. ‎In particular‎, ‎we investigate some results concerning strongly 2-absorbing second submodules‎.

Let [Formula: see text] be a commutative ring with identity, [Formula: see text] be a multiplicatively closed subset of [Formula: see text], and [Formula: see text] be an [Formula: see text]-module. A submodule [Formula: see text] of... more

Let [Formula: see text] be a commutative ring with identity, [Formula: see text] be a multiplicatively closed subset of [Formula: see text], and [Formula: see text] be an [Formula: see text]-module. A submodule [Formula: see text] of [Formula: see text] is called coidempotent if [Formula: see text]. Also, [Formula: see text] is called fully coidempotent if every submodule of [Formula: see text] is coidempotent. In this paper, we introduce the concepts of [Formula: see text]-coidempotent submodules and fully [Formula: see text]-coidempotent [Formula: see text]-modules as generalizations of coidempotent submodules and fully coidempotent [Formula: see text]-modules. We explore some basic properties of these classes of [Formula: see text]-modules.

Rings are built from abelian groups. While ideals (I), special subrings of Rings (R), with Rings, form Quotient Rings (R/I) isomorphic to existing Rings (¯R). Thus, this process creates new Rings. Here, we use this platform to explore... more

Rings are built from abelian groups. While ideals (I), special subrings of Rings (R), with Rings, form Quotient Rings (R/I) isomorphic to existing Rings (¯R). Thus, this process creates new Rings. Here, we use
this platform to explore ideals such as prime, principal and Maximal, based on certain theories to enhance this creativity to our advantage. Examples are considered, examined and explored for applicability of these
theories and creativity of New Rings.

The following theorem is proved: Letr=r(y)>1,s, andtbe non-negative integers. IfRis a lefts-unital ring satisfies the polynomial identity[xy−xsyrxt,x]=0for everyx,y∈R, thenRis commutative. The commutativity of a rights-unital ring... more

The following theorem is proved: Letr=r(y)>1,s, andtbe non-negative integers. IfRis a lefts-unital ring satisfies the polynomial identity[xy−xsyrxt,x]=0for everyx,y∈R, thenRis commutative. The commutativity of a rights-unital ring satisfying the polynomial identity[xy−yrxt,x]=0for allx,y∈R, is also proved.

For any unital commutative ring RRR and for a graph EEE, we identify a maximal commutative subalgebra of the Leavitt path algebra of EEE with coefficients in RRR. Besides we are able to characterize injectivity of representations which... more

For any unital commutative ring RRR and for a graph EEE, we identify a maximal commutative subalgebra of the Leavitt path algebra of EEE with coefficients in RRR. Besides we are able to characterize injectivity of representations which gives a generalization of Cuntz-Krieger uniqueness theorem, and by other hand, to generalize and simplify the result about commutative Leavitt path algebras over fields.

In this paper, we introduce the concept of an mathfrakX\mathfrak{X}mathfrakX-element with respect to an MMM-closed set mathfrakX\mathfrak{X}mathfrakX in multiplicative lattices and study properties of mathfrakX\mathfrak{X}mathfrakX-elements. For a particular MMM-closed subset... more

In this paper, we introduce the concept of an mathfrakX\mathfrak{X}mathfrakX-element with respect to an MMM-closed set mathfrakX\mathfrak{X}mathfrakX in multiplicative lattices and study properties of mathfrakX\mathfrak{X}mathfrakX-elements. For a particular MMM-closed subset mathfrakX\mathfrak{X}mathfrakX, we define the concepts of rrr-elements, nnn-elements and JJJ-elements. These elements generalize the notion of rrr-ideals, nnn-ideals and JJJ-ideals of a commutative ring with identity to multiplicative lattices. In fact, we prove that an ideal III of a commutative ring RRR with identity is a nnn-ideal ($J$-ideal) of RRR if and only if it is an nnn-element ($J$-element) of Id(R)Id(R)Id(R), the ideal lattice of RRR.

Abstract: In this paper we prove; If R is a left quasi-Noetherian ring,then every nil subring is nilpotent). Next we show that a commutative semi-prime quasi-Noetherian ring is Noetherian. Then we study the relationship between left... more

Abstract: In this paper we prove; If R is a left quasi-Noetherian ring,then every nil subring is nilpotent). Next we show that a commutative semi-prime quasi-Noetherian ring is Noetherian. Then we study the relationship between left Quasi-Noetherian and left Quasi-Artinian, in particular we prove that If R is a non-nilpotent left Quasi-Artinian ring. Then any left R-module is left Quasi-Artinian if and only if it is left Quasi-Noetherian. Finally we show that a commutative ring R is Quasi-Artinian if and only ifR is Quasi-Noetherian and every proper prime ideal of R is maximal.

We examine those matrix rings whose entries lie in periodic rings equipped with some additional properties. Specifically, we prove that the famous Diesl’s question of whether or not a ring R is nil-clean implies that the matrix ring Mn(R)... more

We examine those matrix rings whose entries lie in periodic rings equipped
with some additional properties. Specifically, we prove that the famous Diesl’s
question of whether or not a ring R is nil-clean implies that the matrix ring
Mn(R) over R is also nil-clean for all n ≥ 1 paralleling the corresponding
implication for (abelian, local) periodic rings. Besides, we study when the
endomorphism ring E(G) of an abelian group G is periodic. Concretely, we
establish that E(G) is periodic exactly when G is finite as well as we find a
completely necessary and sufficient condition when the endomorphism ring
over an abelian group is strongly m-nil clean for some natural number m
thus refining an “old” result concerning strongly nil-clean endomorphism rings.
Responding to a question when a group ring is periodic, we show that if
R is a right (resp., left) perfect periodic ring and G is a locally finite group,
then the group ring RG is periodic, too. We finally find some criteria under
certain conditions when the tensor product of two periodic algebras over a
commutative ring is again periodic. In addition, some other sorts of rings very
close to periodic rings, namely the so-called weakly periodic rings, are also
investigated.

Etale cohomology is one of the most significant tools introduced by Alexander Grothendieck in algebraic geometry. It addresses limitations in classical cohomology theories, particularly for varieties over fields of positive... more

Etale cohomology is one of the most significant tools introduced by Alexander
Grothendieck in algebraic geometry. It addresses limitations in classical cohomology theories, particularly for varieties over fields of positive characteristic,
like finite fields. This theory enabled deep results, such as the proof of the Weil
conjectures.
Here, I am trying to explain the outlines of ´etale cohomology in a simple
way, making it accessible for all general readers to understand, even without
deep prior knowledge of algebraic geometry or cohomology theories.

In this paper, we present a new construction and decoding of BCH codes over certain rings. Thus, for a nonnegative integer t, let A 0 ⊂ A 1 ⊂ ⋯ ⊂ A t − 1 ⊂ A t be a chain of unitary commutative rings, where each A i is constructed by the... more

In this paper, we present a new construction and decoding of BCH codes over certain rings. Thus, for a nonnegative integer t, let A 0 ⊂ A 1 ⊂ ⋯ ⊂ A t − 1 ⊂ A t be a chain of unitary commutative rings, where each A i is constructed by the direct product of appropriate Galois rings, and its projection to the fields is K 0 ⊂ K 1 ⊂ ⋯ ⊂ K t − 1 ⊂ K t (another chain of unitary commutative rings), where each K i is made by the direct product of corresponding residue fields of given Galois rings. Also, A i ∗ and K i ∗ are the groups of units of A i and K i , respectively. This correspondence presents a construction technique of generator polynomials of the sequence of Bose, Chaudhuri, and Hocquenghem (BCH) codes possessing entries from A i ∗ and K i ∗ for each i, where 0 ≤ i ≤ t. By the construction of BCH codes, we are confined to get the best code rate and error correction capability; however, the proposed contribution offers a choice to opt a worthy BCH code concerning code rate and erro...

Our main purpose is to extend several results of interest that have been proved for modules over integral domains to modules over arbitrary commutative rings RRR with identity. The classical ring of quotients QQQ of RRR will play the role... more

Our main purpose is to extend several results of interest that have been proved for modules over integral domains to modules over arbitrary commutative rings RRR with identity. The classical ring of quotients QQQ of RRR will play the role of the field of quotients when zero-divisors are present. After discussing torsion-freeness and divisibility (Sections 2–3), we study Matlis-cotorsion modules and their roles in two category equivalences (Sections 4–5). These equivalences are established via the same functors as in the domain case, but instead of injective direct sums oplusQ\oplus QoplusQ one has to take the full subcategory of QQQ-modules into consideration. Finally, we prove results on Matlis rings, i.e. on rings for which QQQ has projective dimension 111 (Theorem 6.4).

Abstract. Let m; n ≥ 2 be two positive integers, R a commutative ring with identity and M a unitary R-module. A proper submodule P of M is an (n 􀀀 1; n)-Φm-prime ((n 􀀀 1; n)-weakly prime) submodule if a1; : : : ; an􀀀1 2 R and x 2 M... more

Abstract. Let m; n ≥ 2 be two positive integers, R a commutative ring with identity and M a unitary R-module. A proper submodule P of M is an (n 􀀀 1; n)-Φm-prime ((n 􀀀 1; n)-weakly prime) submodule if a1; : : : ; an􀀀1 2 R and x 2 M together with a1 : : : an􀀀1x 2 Pn(P : M)m􀀀1P (0 = a1 : : : an􀀀1x 2 P) imply a1 : : : ai􀀀1ai+1 : : : an􀀀1x 2 P, for some i 2 f1; : : : ; n􀀀1g or a1:::an􀀀1 2 (P : M). In this paper we study these submodules. Some useful results and examples concerning these types of submodules are given.