\(I^\times C\)-semirings (Q2682773)

A semiring is a generalization of a unitary ring whose additive reduct does not form a group. Therefore, in semirings not every multiplicatively idempotent element is complemented, necessarily. In this paper, the authors introduce and study a new class of semirings in which every multiplicatively idempotent element is complemented. They call this class of semirings, $I^x$ C-semirings, and give some properties of von Neumann regular $I^x$ C-semirings. The main results is a set of: some equivalent criteria on $I^x$ C-semiring $S$, such as: commutative semirings, $Q_{cl}(S)$ of an $I^x$ C-semiring, on $I^x$ C-p.p.semiring, and an almost p.p. semiring etc. The review in some details: The paper contains 3 sections: Introduction, $I^x$ C-semirings, von Neumann regular semirings, and p.p. semirings and almost p.p. semirings. the first section contains some preliminaries, and relevant definitions such as: semiring, complemented element, right multiplicatively cancellable, multiplicatively idempotent, semimodule over a semiring, $S$-congruence relation, Bourne relation, etc. and begins with the definition of $I^x$ C-semiring in section 2. For the purpose to suit the cases, the notions of maximal $S$-semimodule homomorphism as well as that of a projective left semi module spring up just as available in usual ring or module theory most analogously. The various sections contain some lemmas and propositions providing information such as: coincidence of left idempotent cosets, right idempotent identity, coincidence of complement idempotent cosets, an equivalent condition for central idempotent, in a nilpotent-free semiring central property of each multiplicatively idempotent element; a commutative $I^x$ C-semiring with all proper ideals prime is a semifield, an equivalent condition of von Neumann element w.r.t a unit, nsc for projective principal ideal with annihilator, a necessary condition for the coincidence for the annihilator of an element with a right coset, a necessary condition of almost p.p. semiring to be nilpotent free. In the case of a polynomial semiring an equivalent property of mutually vanishing off two polynomials with its corresponding elementwise vanishing characteristics. In a commutative $I^x$ C-semiring S, for $a\in S$, if Ann$(a)=Sf$, $f\in I^x (S)$, then there exist a multiplicatively idempotent e and a nonzero divisor $s\in S$ such that $a=es$. The following are the propositions that the reviewer notes: Proposition. If the $I^x$ C-semiring $S$ is commutative nilpotent-free, then $S$ is von Neumann regular if and only if each finitely generated ideal of $S$ is principal and is generated by a multiplicatively idempotent if and only if every prime ideal of $S$ is maximal if and only if Spec$(S)$ is Hausdorff if and only if every $a\in S$ can be written as a unit times a multiplicatively idempotent. Proposition: The prime character of pseudoprime ideal is a necessary condition of an $I^x$ C-semiring. The propositions deal with when the semiring is a union of zero divisor elements and cancellable elements of it, gives some equivalence between von Neumann character zero divisors. Proposition: In a commutative, $I^x$ C-semiring $S$, every element of $S$ is either nilpotent or von Neumann regular $\leftrightarrow$ either $S$ is von Neumann regular or $S$ is local with maximal ideal nil$(S)$. Propositions: In a commutative, $I^x$ C-semiring $S$, where where each multiplicatively idempotent central, then for $a\in S$, there is $b\in S$ with an integer $n\geq 1$ such that $a^n=a^n ba^n$ if and only if then there is an $e\in I^x$ C$(S)$ with $ea$ von Neumann regular and $e^\perp$ a is nilpotent. Propositions: In a commutative, $I^x$ C-semiring $S$ where each multiplicatively idempotent central: for $a\in S$, there is a $b\in S$ and an integer $n\geq 1$ such that $a^n=a^2b$ if and only if every prime ideal of $S$ is maximal. Proposition: In a commutative, $I^x$ C-semiring $S$: $S$ is p.p. if and only if every element $a\in S$ can be expressed as a product of a non zero divisor and a multiplicatively idempotent if and only if Annihilator of each (r.m.c) element of $S$ is generated by a multiplicatively idempotent. Proposition: every commutative p.p. $I^x$ C-semiring is a nilpotent free semiring, and $S[X]$ is p.p semiring. The main results, in the form of theorems, consist of some equivalent criteria: Theorem: for a commutative $I^x$ C-semiring $S$: For all $s\in S$ there exists a nonzero-divisor $d\in S$ such that $s^2=sd$. if and only if $Q_cl(S)$ is von Neumann regular. Theorem: Let $S$ be a commutative semiring. Then: For every $a\in S$, the ideal $Sa+\mathrm{Ann}(a)$ contains a nonzero divisor. if and only if for every $a\in S$ there is an element $b\in S$ such that a $ab=0$ and $a+b$ is a nonzero-divisor of $S$. Theorem: Let $S$ be a commutative semiring and $Q_{cl}(S)$ be an $I^x$ C-semiring. Then: For every $a\in S$, there is an element $b\in S$ such that $ab=0$ and $a+b$ is a nonzero-divisor of $S$. if and only if $S$ is nilpotent-free and for every $a\in S$, there is an element $b\in S$ such that $\mathrm{Ann}(\mathrm{Ann}(a))=\mathrm{Ann}(b)$ if and only if $Q_{cl}(S)$ is von Neumann regular. Theorem: Let $S$ and $Q_{cl}(S)$ be commutative $I^x$ C-semiring. Then: $S$ is a p.p. semiring if and only if $Q_{cl}(S)$ is von Neumann regular and every multiplicatively idempotent of $Q_{cl}(S)$ belongs to $S$. Theorem: Let $S$ be a commutative $I^x$ C-semiring. Then: $S$ is an almost p.p. semiring if and only if $a,b\in S$ and $ab=0$, then there is a multiplicatively idempotent $e\in S$ such that $a\in eS$ and $b\in e^\perp S$. Let $\{a_i\}_(i=1)^n$ and $\{b_j\}_{(j=1)^k}$ are families of elements of $S$ such that $a_i b_j=0$ for all $1\leq i\leq n$ and $1\leq j\leq k$, then there is a multiplicatively idempotent $e\in S$ such that $a_i=a_ie$ and $b_j=b_je^\perp$ for all $i$ and $j$ Theorem: A commutative $I^x$ C-semiring $S$ is an almost p.p. semiring if and only if $S[X]$ is an almost p.p. semiring.