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Register a new userA Graphical Representation of Rings via Automorphism Groups
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Let $R$ be a commutative ring with identity. We define a graph $Gamma_{aut}(R)$ on $ R$, with vertices elements of $R$, such that any two distinct vertices $x, y$ are adjacent if and only if there exists $sigma in aut$ such that $sigma(x)=y$. The idea is to apply graph theory to study orbit spaces of rings under automorphisms. In this article, we define the notion of a ring of type $n$ for $ngeq 0$ and characterize all rings of type zero. We also characterize local rings $(R,M) $ in which either the subset of units ($ eq 1 $) is connected or the subset $M- {0}$ is connected in $Gamma_{aut}(R)$.
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Let $R$ be a commutative $k-$algebra over a field $k$. Assume $R$ is a noetherian, infinite, integral domain. The group of $k-$automorphisms of $R$,i.e.$Aut_k(R)$ acts in a natural way on $(R-k)$.In the first part of this article, we study the structure of $R$ when the orbit space $(R-k)/Aut_k(R)$ is finite.We note that most of the results, not particularly relevent to fields, in [1,S 2] hold in this case as well. Moreover, we prove that $R$ is a field. In the second part, we study a special case of the Conjecture 2.1 in [1] : If $K/k$ is a non trivial field extension where $k$ is algebraically closed and $mid (K-k)/Aut_k(K) mid = 1$ then $K$ is algebraically closed. In the end, we give an elementary proof of [1,Theorem 1.1] in case $K$ is finitely generated over its prime subfield.
Let $R$ be a commutative ring. We investigate $R$-modules which can be written as emph{finite} sums of {it {second}} $R$-submodules (we call them emph{second representable}). We provide sufficient conditions for an $R$-module $M$ to be have a (minimal) second presentation, in particular within the class of lifting modules. Moreover, we investigate the class of (emph{main}) emph{second attached prime ideals} related to a module with such a presentation.
Let $R$ be a commutative ring with identity. In this paper, we introduce the concept of weakly $1$-absorbing prime ideals which is a generalization of weakly prime ideals. A proper ideal $I$ of $R$ is called weakly $1$-absorbing prime if for all nonunit elements $a,b,c in R$ such that $0 eq abc in I$, then either $ab in I$ or $c in I$. A number of results concerning weakly $1$-absorbing prime ideals and examples of weakly $1$-absorbing prime ideals are given. It is proved that if $I$ is a weakly $1$-absorbing prime ideal of a ring $R$ and $0 eq I_1I_2I_3 subseteq I$ for some ideals $I_1, I_2, I_3$ of $R$ such that $I$ is free triple-zero with respect to $I_1I_2I_3$, then $ I_1I_2 subseteq I$ or $I_3subseteq I$. Among other things, it is shown that if $I$ is a weakly $1$-absorbing prime ideal of $R$ that is not $1$-absorbing prime, then $I^3 = 0$. Moreover, weakly $1$-absorbing prime ideals of PIDs and Dedekind domains are characterized. Finally, we investigate commutative rings with the property that all proper ideals are weakly $1$-absorbing primes.
The behaviour under coarsening functors of simple, entire, or reduced graded rings, of free graded modules over principal graded rings, of superfluous monomorphisms and of homological dimensions of graded modules, as well as adjoints of degree restriction functors, are investigated.
Let $M$ be an $R$-module and $c$ the function from $M$ to the ideals of $R$ defined by $c(x) = cap lbrace I colon I text{is an ideal of} R text{and} x in IM rbrace $. $M$ is said to be a content $R$-module if $x in c(x)M $, for all $x in M$. $B$ is called a content $R$-algebra, if it is a faithfully flat and content $R$-module and it satisfies the Dedekind-Mertens content formula. In this article, we prove some new results for content modules and algebras by using ideal theoretic methods.
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