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| Paper IPM / M / 16184 |
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| Abstract: | |
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Let R denote a commutative Noetherian ring and I an ideal of R. The
concept of quintessential sequences over zero ideal was introduced by McAdam and
Ratli [8]. They showed that these sequences enjoy many of the basic properties of
asymptotic sequences over zero ideal. It was shown, that quintessential sequences over
ideals I ̸= (0)R are not a good analogue of asymptotic sequences over I ̸= (0)R. By
making use of the new concept of quintessential grade of an ideal over another ideal,
we show that there exists a class of ideals I for which quintessential sequences over I
are an excellent analogue of asymptotic sequences over I. Also, we give more results
on quintessential sequences over an ideal and derive generalizations of some McAdam-
Ratli 's results [8, 13].
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