The soft Jacobson radical of a commutative ring

In this paper, the notion of the soft Jacobson radical of a ring is defined. A relationship between the soft Jacobson radical of a ring and Jacobson semisimple ring is established. Some properties of this notion have been studied under homomorphism.

Some variations of projectivity

We prove that a ring [Formula: see text] has a module [Formula: see text] whose domain of projectivity consists of only some injective modules if and only if [Formula: see text] is a right noetherian right [Formula: see text]-ring. Also, we consider modules which are projective relative only to a subclass of max modules. Such modules are called max-poor modules. In a recent paper Holston et al. showed that every ring has a p-poor module (that is a module whose projectivity domain consists precisely of the semisimple modules). So every ring has a max-poor module. The structure of all max-poor abelian groups is completely determined. Examples of rings having a max-poor module which is neither projective nor p-poor are provided. We prove that the class of max-poor [Formula: see text]-modules is closed under direct summands if and only if [Formula: see text] is a right Bass ring. A ring [Formula: see text] is said to have no right max-p-middle class if every right [Formula: see text]-module is either projective or max-poor. It is shown that if a commutative noetherian ring [Formula: see text] has no right max-p-middle class, then [Formula: see text] is the ring direct sum of a semisimple ring [Formula: see text] and a ring [Formula: see text] which is either zero or an artinian ring or a one-dimensional local noetherian integral domain such that the quotient field [Formula: see text] of [Formula: see text] has a proper [Formula: see text]-submodule which is not complete in its [Formula: see text]-topology. Then we show that a commutative noetherian hereditary ring [Formula: see text] has no right max-p-middle class if and only if [Formula: see text] is a semisimple ring.

Isomorphism between Endomorphism Rings of Modules over A Semisimple Ring

Our question is what ring R which all modules over R are determined, up to isomorphism, by their endomorphism rings? Examples of this ring are division ring and simple Artinian ring. Any semi simple ring does not satisfy this property. We construct a semi simple ring R but R is not a simple Artinian ring which all modules over R are determined, up to isomorphism, by their endomorphism rings.

Some algebraic and homological properties of a family of quotients of the Rees algebra

Let [Formula: see text] be a commutative ring and let [Formula: see text] be a proper ideal of [Formula: see text]. In this paper, we study some algebraic and homological properties of a family of rings [Formula: see text], with [Formula: see text], that are obtained as quotients of the Rees algebra associated with the ring [Formula: see text] and the ideal [Formula: see text]. Specially, we study when [Formula: see text] is a von Neumann regular ring, a semisimple ring and a Gaussian ring. Also, we study the classical global and weak global dimensions of [Formula: see text]. Finally, we investigate some homological properties of [Formula: see text]-modules and we show that [Formula: see text] and [Formula: see text] are Gorenstein projective [Formula: see text]-modules, provided some special conditions.

Small potential counterexamples to the pure semisimplicity conjecture

The pure semisimplicity conjecture or pssc states that every left pure semisimple ring has finite representation type. Let [Formula: see text] be division rings, and assume we identify conditions on a [Formula: see text]-[Formula: see text]-bimodule [Formula: see text] which are sufficient to make the triangular matrix ring [Formula: see text] into a left pure semisimple ring which is not of finite representation type. It is then said that those conditions yield a potential counterexample to the pssc. Simson [17–20] gave several such conditions in terms of the sequence of the left dimensions of the left dual bimodules of [Formula: see text]. In this paper, conditions with the same purpose are given in terms of the continued fraction attached to [Formula: see text], and also through arithmetical properties of a division ring extension [Formula: see text].

On Baer–Kaplansky Classes of Modules

We are interested in studying when the class of local modules is Baer–Kaplansky. We provide an example showing that even over a commutative semisimple ring R, we can find two non-isomorphic simple R-modules S1 and S2 such that the rings EndR(S1) and EndR(S2) are isomorphic. We show that over any ring R, the class of semisimple R-modules is Baer–Kaplansky if and only if so is the class of simple R-modules.

Filtration and Dubrovin valuation on matrix ring

In this paper, we show if [Formula: see text] is a filtered ring then we can define a filtration on [Formula: see text] which induced by a filtration on [Formula: see text], and we prove some properties and relations for [Formula: see text]. Also, we show if [Formula: see text] is a filtered ring then we can define a filtration on [Formula: see text], and we prove some properties and relation for [Formula: see text]. Later we show that, there exist a Dubrovin valuation on [Formula: see text], if [Formula: see text] is a semisimple ring.

A note on rings with the summand sum property

A ring R is called right SSP (SIP) if the sum (intersection) of any two direct summands of RR is also a direct summand. Left SSP (SIP) rings are defined similarly. There are several interesting results on rings with SSP. For example, R is right SSP if and only if R is left SSP, and R is a von Neumann regular ring if and only if Mn(R) is SSP for some n > 1. It is shown that R is a semisimple ring if and only if the column finite matrix ring ℂFMℕ(R) is SSP, where ℕ is the set of natural numbers. Some known results are proved in an easy way through idempotents of rings. Moreover, some new results on SSP rings are given.