# EXTENSIONS OF EXTENDED SYMMETRIC RINGS

• Published : 2007.11.30
• 53 6

#### Abstract

An endomorphism ${\alpha}$ of a ring R is called right(left) symmetric if whenever abc=0 for a, b, c ${\in}$ R, $ac{\alpha}(b)=0({\alpha}(b)ac=0)$. A ring R is called right(left) ${\alpha}-symmetric$ if there exists a right(left) symmetric endomorphism ${\alpha}$ of R. The notion of an ${\alpha}-symmetric$ ring is a generalization of ${\alpha}-rigid$ rings as well as an extension of symmetric rings. We study characterizations of ${\alpha}-symmetric$ rings and their related properties including extensions. The relationship between ${\alpha}-symmetric$ rings and(extended) Armendariz rings is also investigated, consequently several known results relating to ${\alpha}-rigid$ and symmetric rings can be obtained as corollaries of our results.

#### Keywords

reduced rings;symmetric rings;(extended) Armendariz rings

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#### Cited by

1. On Extensions of Right Symmetric Rings without Identity vol.04, pp.12, 2014, https://doi.org/10.4236/apm.2014.412075