Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Weakly $I$-clean rings   In this article, we introduce the concept of weakly $I$-clean ring, for any
ideal $I$ of a ring $R$. We show that, for an ideal $I$ of a ring $R$, $R$ is
uniquely weakly $I$-clean if and only if $R/I$ is semi boolean and idempotents
can be lifted uniquely weakly modulo $I$ if and only if for each $a\in R$,
there exists a central idempotent $e\in R$ such that either $a-e\in I$ or
$a+e\in I$ and $I$ is idempotent free. As a corollary, we characterize weakly
$J$-clean ring. Also we study various properties of weakly $I$-clean ring.

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The categories of this paper are: nucl-th
32
The categories of this paper are: atom-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: patt-sol
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-th
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The categories of this paper are: stat
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The categories of this paper are: q-alg
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The categories of this paper are: eess
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The categories of this paper are: quant-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: funct-an
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-ph
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The categories of this paper are: adap-org
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The categories of this paper are: nlin