Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Weakly $r$-clean rings and weakly $\star$-clean rings   Motivated by the concept of weakly clean rings, we introduce the concept of
weakly $r$-clean rings. We define an element $x$ of a ring $R$ as weakly
$r$-clean if it can be expressed as $x=r+e$ or $x=r-e$ where $e$ is an
idempotent and $r$ is a regular element of $R$. If all the elements of $R$ are
weakly $r$-clean then $R$ is called a weakly $r$-clean ring. We discuss some of
its properties in this article. Also we generalise this concept of weakly
$r$-clean ring to weakly $g(x)$-$r$-clean ring, where $g(x) \in C(R)[x]$ and
$C(R)$ is the centre of the ring $R$. Finally we introduce the concept of
weakly $\star$-clean and $\star$-$r$-clean ring and discuss some of their
properties.

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The categories of this paper are: stat
33
The categories of this paper are: plasm-ph
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The categories of this paper are: math-ph
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The categories of this paper are: supr-con
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The categories of this paper are: q-fin
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The categories of this paper are: q-bio
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The categories of this paper are: econ
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The categories of this paper are: funct-an
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The categories of this paper are: cond-mat
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The categories of this paper are: bayes-an
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The categories of this paper are: mtrl-th
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The categories of this paper are: astro-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: eess
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The categories of this paper are: comp-gas
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The categories of this paper are: chao-dyn