Let I be any index set. We consider the Banach algebra Ce + l(I) with the Hadamard product, and prove that its Bass and topological stable ranks are both equal to 1. We also characterize divisors, maximal ideals, closed ideals and closed principal ideals. For I = N we also characterize all prime z-ideals in this Banach algebra.

Let $G$ be a finitely generated abelian group and $M$ be a $G$-graded $A$-module. In general, $G$-associated prime ideals to $M$ may not exist. In this paper, we introduce the concept of $G$-attached prime ideals to $M$ as a generalization of $G$-associated prime ideals which gives a connection between certain $G$-prime ideals and $G$-graded modules over a (not necessarily $G$-graded Noetherian...

We investigate the ideal structures of the C∗-algebras arising from topological graphs. We give the complete description of ideals of such C∗-algebras which are invariant under the so-called gauge action, and give the condition on topological graphs so that all ideals are invariant under the gauge action. We get conditions for our C∗algebras to be simple, prime or primitive. We completely deter...

1 Definitions Definition 1. A graded ring is a ring S together with a set of subgroups Sd, d ≥ 0 such that S = ⊕ d≥0 Sd as an abelian group, and st ∈ Sd+e for all s ∈ Sd, t ∈ Se. One can prove that 1 ∈ S0 and if S is a domain then any unit of S also belongs to S0. A homogenous ideal of S is an ideal a with the property that for any f ∈ a we also have fd ∈ a for all d ≥ 0. A morphism of graded r...

In a commutative f-ring, an 1-ideal I is called pseudoprime if ab = 0 implies a E I or b E I, and is called square dominated if for every a E I, lal < x2 for some x E A such that x2 E I. Several characterizations of pseudoprime 1-ideals are given in the class of commutative semiprime frings in which minimal prime 1-ideals are square dominated. It is shown that the hypothesis imposed on the f-ri...

In this paper, we introduce a new generalization of weakly prime ideals called $I$-prime. Suppose $R$ is a commutative ring with identity and $I$ a fixed ideal of $R$. A proper ideal $P$ of $R$ is $I$-prime if for $a, b in R$ with $ab in P-IP$ implies either $a in P$ or $b in P$. We give some characterizations of $I$-prime ideals and study some of its properties.  Moreover, we give conditions  ...

The rings considered in this article are commutative with identity $1neq 0$. By a proper ideal of a ring $R$,  we mean an ideal $I$ of $R$ such that $Ineq R$.  We say that a proper ideal $I$ of a ring $R$ is a  maximal non-prime ideal if $I$ is not a prime ideal of $R$ but any proper ideal $A$ of $R$ with $ Isubseteq A$ and $Ineq A$ is a prime ideal. That is, among all the proper ideals of $R$,...

It is well-known that the sum of two $z$-ideals in $C(X)$ is either $C(X)$ or a $z$-ideal. The main aim of this paper is to study the sum of strongly $z$-ideals in ${mathcal{R}} L$, the ring of real-valued continuous functions on a frame $L$. For every ideal $I$ in ${mathcal{R}} L$, we introduce the biggest strongly $z$-ideal included in $I$ and the smallest strongly $z$-ideal containing ...

the rings considered in this article are commutative with identity $1neq 0$. by a proper ideal of a ring $r$,  we mean an ideal $i$ of $r$ such that $ineq r$.  we say that a proper ideal $i$ of a ring $r$ is a  maximal non-prime ideal if $i$ is not a prime ideal of $r$ but any proper ideal $a$ of $r$ with $ isubseteq a$ and $ineq a$ is a prime ideal. that is, among all the proper ideals of $r$,...

Let N be a near-ring. In this paper, we associate a graph corresponding to the 3-prime radical I of N , denoted by ΓI(N). Further we obtain certain topological properties of Spec(N), the spectrum of 3-prime ideals of N and graph theoretic properties of ΓI(N). Using these properties, we discuss dominating sets and connected dominating sets of ΓI(N).