JP Journal of Algebra, Number Theory and Applications
Volume 39, Issue 1, Pages 11 - 20
(February 2017) http://dx.doi.org/10.17654/NT039010011 |
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-MODULES AND A SPECIAL CLASS OF MODULES DETERMINED BY THE ESSENTIAL CLOSURE OF THE CLASS OF ALL *-RINGS
Puguh Wahyu Prasetyo, Indah Emilia Wijayanti and Halina France-Jackson
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Abstract: A ring A is called a *-ring if A is a prime ring and A has no nonzero proper prime homomorphic image. The *-ring was introduced by Korolczuk in 1981. Since *-rings have an important role in radical theory of rings, the properties of *-ring have been being investigated intensively. Since every ring can be viewed as a module over itself, the generalization of *-ring into module theory is an interesting investigation. We would like to present the generalization of *-rings in module theory named -modules. An A-module M is called a -module if M is a prime A-module and M has no nonzero proper prime submodule. According to the result of our investigation, we show that every *-ring is a -module over itself. Furthermore, let A be a ring, let M be an A-module, and let I be an ideal of A with where We show that M is a -module over A if and only if M is a -module over On the other hand, the essential closure of the class of all *-rings is a special class of rings. As the last result of our investigation, we present the special class of modules determined by |
Keywords and phrases: *-ring, homomorphic image, special class of rings, special class of modules, prime module. |
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