Journal Name:
- European Journal of Pure and Applied Mathematics
| Author Name |
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Abstract (2. Language):
Let R be a ring and an endomorphism of a ring R. Recall that R is said to be a (∗)-ring
if a(a) ∈ P(R) implies a ∈ P(R) for a ∈ R, where P(R) is the prime radical of R. We also recall that R
is said to be a weak -rigid ring if a(a) ∈ N(R) if and only if a ∈ N(R) for a ∈ R, where N(R) is the
set of nilpotent elements of R.
In this paper we give a relation between a (∗)-ring and a weak -rigid ring. We also give a necessary
and sufficient condition for a Noetherian ring to be a weak -rigid ring. Let be an endomorphism
of a ring R and a -derivation of R such that ((a)) = ((a)) for all a ∈ R. Then can be
extended to an endomorphism (say ) of R[x;,] and can be extended to a -derivation (say )
of R[x;,]. With this we show that if R is a 2-primal commutative Noetherian ring which is also
an algebra over Q (where Q is the field of rational numbers), is an automorphism of R and a
-derivation of R such that ((a)) = ((a)) for all a ∈ R, then R is a weak -rigid ring implies that
R[x;,] is a weak -rigid ring.
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