Right Gaussian rings and related topics

Prüfer domains are commutative domains in which every non -zero finitely generated ideal is invertible. Since such domains play a central role in multiplicative ideal theory, any equivalent condition to the Prüfer domain notion is of great interest. It occurs that the class of Prüfer domains is equivalent to other classes which are investigated in theory of commutative rings (see [22D. Namely, for commutative rings the following classes are equivalent:(1) Semihereditary domains.(2) Domains which have weak dimension less or equal to one.(3) Distributive domains.(4) Gaussian domains.(5) Prüfer domains.Many authors have studied so called Prüfer rings which are a generalization of notion of Prüfer domains to the case of commutative rings with zero divisors. In this context there are investigated the following classes of commutative rings:(I) Semihereditary rings.(II) Rings which have weak dimension less or equal to one.(III) Distributive rings.(IV) Gaussian rings.(V) Prüfer rings.Recently the main stress in the area is focused on Gaussian rings (e.g. see [8] or [22]). In [22] S. Glaz showed that we have (I) —» (II) —» (III) —» (IV) —» (V) and no one of these implications can be replaced by the equivalence.In this thesis the notion of a Gaussian ring is extended to the noncommutative setting by introducing a new class of rings which are called right Gaussian rings. We investigate the relations with noncommutative analogs of classes (I), (II), (III), (IV), and in some cases (V). Moreover, we study some related subjects which naturally occur during our research concerning right Gaussian rings.In Chapter 2 we recall some facts regarding right distributive rings, and define right Gaussian rings. Moreover, we study basic properties of right Gaussian rings. We also present results about the connection between the above classes of rings.Chapter 3 includes an investigation about right Gaussian skew power series rings. We will give an extension to the noncommutative case of a well -known result by Anderson and Camillo (see [2, Theorem 17]).In Chapter 4 we define skew generalized power series rings and for positively ordered monoids we describe those of above which are right Gaussian.It occurs that for a right Gaussian ring a ring of quotients may not exist, and even when it exists, it need not be right Gaussian. We study relevant these issues formulate Chapter 5.In Chapter 6 we consider a class of homomorphie images of a polynomial ring R[x] and give the necessary and sufficient conditions for a ring R under which these images are right Gaussian.In Chapter 7 we make an effort to establish what kind of relations hold among right Gaussian rings, right Prüfer rings and some other classes of noncommutative rings.Right Gaussian rings are exactly right duo Armendariz rings. This fact is a reason to take on Armendariz rings in detail, which we do in Chapter 8.The final chapter contains investigations about some subclasses of unique product monoids which appear naturally in Chapter 8

Semicommutativity of the rings relative to prime radical

summary:In this paper, we introduce a new kind of rings that behave like semicommutative rings, but satisfy yet more known results. This kind of rings is called $P$-semicommutative. We prove that a ring $R$ is $P$-semicommutative if and only if $R[x]$ is $P$-semicommutative if and only if $R[x, x^{-1}]$ is $P$-semicommutative. Also, if $R[[x]]$ is $P$-semicommutative, then $R$ is $P$-semicommutative. The converse holds provided that $P(R)$ is nilpotent and $R$ is power serieswise Armendariz. For each positive integer $n$, $R$ is $P$-semicommutative if and only if $T_n(R)$ is $P$-semicommutative. For a ring $R$ of bounded index $2$ and a central nilpotent element $s$, $R$ is $P$-semicommutative if and only if $K_s(R)$ is $P$-semicommutative. If $T$ is the ring of a Morita context $(A,B,M,N,\psi,\varphi)$ with zero pairings, then $T$ is $P$-semicommutative if and only if $A$ and $B$ are $P$-semicommutative. Many classes of such rings are constructed as well. We also show that the notions of clean rings and exchange rings coincide for $P$-semicommutative rings

NonCommutative Rings and their Applications, IV ABSTRACTS Checkable Codes from Group Algebras to Group Rings

Abstract A code over a group ring is defined to be a submodule of that group ring. For a code C over a group ring RG, C is said to be checkable if there is v ∈ RG such that C = {x ∈ RG : xv = 0}. In [1], Jitman et al. introduced the notion of code-checkable group ring. We say that a group ring RG is code-checkable if every ideal in RG is a checkable code. In their paper, Jitman et al. gave a necessary and sufficient condition for the group ring FG, when F is a finite field and G is a finite abelian group, to be codecheckable. In this paper, we generalize this result for RG, when R is a finite commutative semisimple ring and G is any finite group. Our main result states that: Given a finite commutative semisimple ring R and a finite group G, the group ring RG is code-checkable if and only if G is π -by-cyclic π; where π is the set of noninvertible primes in R