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Vietnam
Journal of Mathematics 35:1(2007)
11-19
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Some
Examples of ACS-Rings
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Qingyi Zeng
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Abstract. A ring R is called a right ACS-ring if the annihilator of any element
in R is essential in a direct
summand of R. In this note we
will exhibit some elementary but important examples of ACS-rings. Let R be a reduced ring, then R is a right ACS-ring if and only if
R[x] is a right ACS-ring. Let R
be an a-rigid ring. Then R is a right ACS-ring if and only if
the Ore extension R[x;a] is a right ACS-ring. A
counterexample is given to show that the upper matrix ring Tn(R) over a right ACS-ring R
need not be a right ACS-ring.
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2000
Mathematics Subject Classification: 16E50, 16N99.
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Keywords: ACS-rings; annihilators;
idempotents; essential; extensions of rings.
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Established
by Vietnam Academy of Science and Technology & Vietnam Mathematical
Society
Published by
Springer since January 2013
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