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Vietnam Journal of Mathematics 35:1(2007) 11-19

 Some Examples of ACS-Rings

Qingyi Zeng

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.

 

2000 Mathematics Subject Classification: 16E50, 16N99.

Keywords: ACS-rings; annihilators; idempotents; essential; extensions of rings.

 

 

 

 

 

 

 

 

 

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