Study of Noetherian Γ-Semirings via Operator Semirings
TL;DR: In this paper, the authors studied Noetherian Γ-semirings and obtained Cohen's theorem for weak primary decomposition for a special class of Γ semirings.
Abstract: In this paper we study Noetherian Γ-semirings and obtain Cohen’s theorem for a special class of Γ-semirings Weak primary decomposition theorem for a particular type of Γ-semirings is also obtained
Citations
More filters
01 Jan 2005
TL;DR: In this paper, the notions of Jacobson radical of a semiring and semisimple semiring were introduced and characterized via operator semirings, and a number of characterizations of semimodule semiring was obtained.
Abstract: We introduce the notions of Jacobson radical of a semiring and semisimple semiring and characterize them via operator semirings. AMS Mathematics Subject Classification (2000): 16Y60, 16Y99, 20N10 characterizations. We obtain the relation between the Jacobson radical of a semiring S and that of the right operator semiring R of S, which we use to obtain characterizations of Jacobson radical of a semiring analogous to familiar results of the ring theory and semiring theory, (5). Then, with the help of the notion of subdirect sum of semiring, introduced at the outset in a similar way to that in ring, (6), and using the result that "a semiring S is semisimple if and only if its right operator semi- ring R is semisimple", a number of characterizations of semisimple semiring is obtained. For preliminaries of semirings, semirings, operator semirings of a semi- ring and rings we refer to (4), (9), (1), (6) and references therein. Throughout this paper a semiring is assumed to be with zero, the left unity, the right unity. It is also assumed that a S semimodule is additively cancellative.
38 citations
Cites background from "Study of Noetherian Γ-Semirings via..."
...We first observe that P ∗ ′ is the right operator semiring of P ([3]) considered as a Γ−semiring (in fact R = S∗)....
[...]
01 Jan 2010
TL;DR: In this paper, the concepts of h-prime ideals and h-semiprime ideals in semirings and Γ-semirings are introduced so that their properties are studied.
Abstract: The concepts of h-prime ideals and h-semiprime ideals in semirings and Γ-semirings are introduced so that their properties are studied. In particular, the relationships between Γ-semirings and its related operator semirings are described in terms of the h-closure; the h-prime and h-semiprime ideals. These results will be used to obtain some other new results such as the inclusion preserving bijections between the h-prime (or h-semiprime) ideals of a Γ-semiring and its related operator semirings. Moreover, the h-regularity and the H-Noetherian Γ-semirings will be characterized. Some recent results given by T. K. Dutta and S. K. Sardar in semiprime ideals and irreducible ideals of a Γ- Semirings and extended and generlized. Mathematics Subject Classification: 16Y60, 16Y99, 20N10
4 citations
Cites background or methods from "Study of Noetherian Γ-Semirings via..."
...We first prove the following crucial lemmas because it have already been shown in [4] that each of the functions “∗, ∗, + ,+ ” [2] is commutative with the operation H ....
[...]
...Let S be H-Noetherian and P1 ⊆ P2 ⊆ ... an ascending chain of ideals of L....
[...]
...Mathematics Subject Classification: 16Y60, 16Y99, 20N10 Keywords: h-prime ideals; h-semiprime ideals; Γ-semiringss; Operator semirings; H-Noetherian Γ-semirings ; h-regularity...
[...]
...Moreover, the h-regularity and the H-Noetherian Γ-semirings will be characterized....
[...]
...The following Theorems of a Γ-semiring was first stated and proved by Dutta and Sardar in [4]....
[...]
TL;DR: In this paper, it was shown that the notion of primary ideals in Γ− semirings inherits most of the essential properties of the primary ideals of commutative semiring.
Abstract: In this paper, we study some results on the ideal theory of commutative Γ− semirings analogues to commutative semirings . In particular, Q-ideals , maximal ideals, primary ideals and radical ideals of commutative Γ− semiring are investigated . Furthermore we make an intensive examination of the notions of maximal ideal and local Γ− semirings. It is shown that the notion of primary ideals in Γ− semirings inherits most of essential properties of primary ideals of commutative semirings. Mathematics Subject Classification: 16Y60, 16Y99
Cites methods from "Study of Noetherian Γ-Semirings via..."
...[6] An ideal P of Γ− semiring S is said to be primary ideal of S if xαy ∈ P, α ∈ Γ, x, y ∈ S then x ∈ P or (yβ)y ∈ P , for all β ∈ Γ for a positive integer n....
[...]
...Noetherian Γ− semirings, Cohen’s theorem for a special class of Γ− semiring and weak primary decomposition theorem for a particular type of Γ− semirings were obtained by them [6]....
[...]
TL;DR: In this article , it was shown that if {Ai,i∈Λ] is a family of strong ideals of a Γ-semiring R, then ∩i, ∪i ∈ ΛAi are strong.
Abstract: In this paper, first we set up some properties of strong ideals of R, which are analogous to the results in semirings and prove that if {Ai}i∈Λ is a family of strong ideals of a Γ– semiring R then ∩i∈ΛAi, ∪i∈ΛAi are strong and ∪i∈ΛAi = Σi∈ΛAi. Finally, we sum up the semiring hypothetical consequences of non-commutative semirings concerning the k-primary ideals toΓ– semirings.
References
More filters
Book•
01 Jan 1992TL;DR: Eventually, you will entirely discover a new experience and talent by spending more cash, which will guide you to understand more just about the globe, experience, some places, when history, amusement, and a lot more.
Abstract: Eventually, you will entirely discover a new experience and talent by spending more cash. yet when? do you agree to that you require to get those all needs afterward having significantly cash? Why don't you attempt to get something basic in the beginning? That's something that will guide you to understand even more just about the globe, experience, some places, when history, amusement, and a lot more?
278 citations
Book•
01 Jan 1995
TL;DR: In this paper, the structure theory of semigroups is examined and structure theorems on regular and commutative semiigroups are introduced; see Section 2.1.
Abstract: This work offers concise coverage of the structure theory of semigroups. It examines constructions and descriptions of semigroups and emphasizes finite, commutative, regular and inverse semigroups. Many structure theorems on regular and commutative semigroups are introduced.;College or university bookstores may order five or more copies at a special student price which is available upon request from Marcel Dekker, Inc.
175 citations
TL;DR: In this article, the class of semirings S for which the semiirings of square matrices Mn(S) over S are (left) k-artinian is characterized.
Abstract: In this paper we characterize the class of semirings S for which the semirings of square matrices Mn(S) over S are (left) k-artinian. Also an analogue of the Hilbert basis theorem for semirings is obtained.
8 citations
"Study of Noetherian Γ-Semirings via..." refers methods in this paper
...Now we have the following characterization of right h-(k-)Noetherian G-semiring which is analogous to that of semiring theory ([ 9 ]):...
[...]