An error has been kindly pointed out to the author by Prof. K. Veradarajan and hence one needs to split original Theorem 5.1 into two theorems.
Theorem 5.1
Let R be a ring and x an element in R. If x is an element of R and x = a + 1, where a is periodic, then x is clean.
Theorem 5.2
Let R be an algebra over a field F of characteristic zero. If x is an element of R and if x = a − 1, where a is periodic, then x is clean.
Corollary 5.1
Let R be an algebra over a field F of characteristic zero. If each element in R is a sum of periodic element and 1 and −1, then R is clean.
Notes
Communicated by K. Misra.
REFERENCES
- Ye , Y. ( 2003 ). Semiclean rings . Comm. Alg. 31 ( 11 ): 5609 – 5625 . [CSA]
- Communicated by K. Misra.