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Decomposition of Elements of a Right Self Injective Ring, III

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Feroz Siddique, Saint Louis University

What
  • Algebra Seminar
When Thu, Sep 20, 2012
from 02:10 PM to 03:00 PM
Where Ritter Hall 134
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Abstract. A ring  R  is called 2-good if each element in  R is a sum of two units.  It was proved independently by both Wolfson and Zelinsky that the ring of linear transformations of a vector space  V  over a division ring  D  is 2-good except when  V  is one-dimensional over  ℤ₂.  A ring  R  is called right self-injective if each right R-homomorphism from any right ideal of  R  to  R  can be extended to an endomorphism of  R.  

As the ring of linear transformations is a right self-injective ring the result of Wolfson and Zelinsky generated quite a bit of attention towards understanding which right self-injective rings are 2-good.  Khurana and Srivastava completely characterized right self injective 2-good rings by proving that a right self injective ring is 2-good if and only if it has no factor isomorphic to  ℤ₂.  

 A ring  R  is called twin-good if for each element  a ∊ R  there exists a unit  u ∊ R  such that both  a+u  and  a-u  are units. Clearly every twin good ring is 2-good but converse need not be true.  We prove (jointly with Professor Srivastava), that if R  is a right self-injective ring then  R  is twin good if and only if  R  has no factor isomorphic to  ℤ₂  or  ℤ₃.  

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