Class Of Simple Rings Research Articles

On Unit Fine Rings

A nonzero element a in a unital ring R is called unit fine if there is a unit u in R such that ua is fine (i.e. a sum of a unit and a nilpotent). A ring all whose elements are unit fine is called accordingly. This turns out to be a new class of simple rings, including the class of fine rings (and so the class of simple Artinian rings). The paper studies unit fine elements and rings. For rings such that 1 is a sum of two units, it is proved that matrix rings over unit fine rings, are unit fine.

Generalized fine rings

As introduced by Cǎlugǎreanu and Lam in [G. Cǎlugǎreanu and T. Y. Lam, Fine rings: a new class of simple rings, J. Algebra Appl. 15(9) (2016) 1650173, 18 pp.], a fine ring is a ring whose every nonzero element is the sum of a unit and a nilpotent. As a natural generalization of fine rings, a ring is called a generalized fine ring if every element not in the Jacobson radical is the sum of a unit and a nilpotent. Here some known results on fine rings are extended to generalized fine rings. A notable result states that matrix rings over generalized fine rings are generalized fine, extending the important result in [G. Cǎlugǎreanu and T. Y. Lam, Fine rings: a new class of simple rings, J. Algebra Appl. 15(9) (2016) 1650173, 18 pp.] that matrix rings over fine rings are fine.

Rings with fine idempotents

An idempotent in a ring is called fine (see G. Călugăreanu and T. Y. Lam, Fine rings: A new class of simple rings, J. Algebra Appl. 15(9) (2016) 18) if it is a sum of a nilpotent and a unit. A ring is called an idempotent-fine ring (briefly, an [Formula: see text] ring) if all its nonzero idempotents are fine. In this paper, the properties of [Formula: see text] rings are studied. A notable result is proved: The diagonal idempotents [Formula: see text] ([Formula: see text]) are fine in the matrix ring [Formula: see text] for any unital ring [Formula: see text] and any positive integer [Formula: see text]. This yields many classes of rings over which matrix rings are [Formula: see text].

Radicals related to the Brown-McCoy radical in some varieties of algebras

AbstractThe Brown-McCoy radical is the upper radical defined by the class of simple rings with identities. For associative or alternative rings the Brown-McCoy radical is hereditary, and its semi-simple class consists of all subdirect products of simple rings with identities. In this paper we present some classes of simple non-associative algebras whose upper radicals behave similarly. Classifications are then obtained of ‘most’ semi-simple radical classes of (γ, δ) and right alternative rings.

On a class of simple rings

MathematikaVolume 5, Issue 2 p. 103-117 Research Article On a class of simple rings P. M. Cohn, P. M. Cohn The University, Manchester 13Search for more papers by this author P. M. Cohn, P. M. Cohn The University, Manchester 13Search for more papers by this author First published: 01 December 1958 https://doi.org/10.1112/S002557930000142XCitations: 8AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume5, Issue2December 1958Pages 103-117 RelatedInformation