Conference abstracts

Session S05 - Rings and Algebras

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Semiclean Rings

Elen Deise Barbosa

Universidade Federal da Bahia, Brasil   -

A ring $R$ with unity is said to be clean if every element in the ring can be written as the sum of a unit and an idempotent of the ring. These rings were introduced by Nicholson, \cite{1}, in his study of lifting idempotents and exchange rings. The division rings, boolean rings and local rings are examples of clean rings.

In the article \cite{6}, a new class of rings is defined; semiclean rings . A ring $R$ with unity is called semiclean if, every $x \in R$, $x= u + a$ with $u \in \mathcal{U}(R)$ where $a$ is periodic element, i.e., $a^k=a^l$ with $k,z \in \mathbb{Z}$ and $k\neq z$. Therefore, every semiclean ring is a clean ring, because the idempotents elements of ring are periodics. Nicholson e Han, \cite{5}, demonstrated that group ring $Z_{(7)}C_3$ is not a clean ring. Yuanqing Ye showed, in the article \cite{6}, that the group ring $Z_{(p)}C_3$ is an semiclean ring. This result assures that the two classes, clean and semiclean, are different.

Motivated by the article \cite{6}, we intend to investigate if the Yuanging Ye's demonstration can be generalized, as in the cases $Z_{(11)}C_5$ and $Z_{(p)}C_5$, in search of a possible answer about the ring $Z_{(p)}C_q$ with $p$ and $q$ relatively primers.

Joint work with Elen Deise Assis Barbosa(Universidade Federal da Bahia, Brasil).

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