Generalization of Nilpotency of Ring Elements to Module Elements

Corrigendum

“…Propositions 2.3 and 2.4 of [1] are valid, when the above definition of a strongly nilpotent element of a module is appropriately used in the given proofs. Furthermore, Theorem 3.1 and Corollaries 3.1 and 3.2 of [1] are valid if on top of the given conditions the module is uniserial.…”

“…Furthermore, Theorem 3.1 and Corollaries 3.1 and 3.2 of [1] are valid if on top of the given conditions the module is uniserial. A module is uniserial if its submodules are totally ordered by inclusion.…”

“…

The abstract of [1] should read as follows: We define nilpotent and strongly nilpotent elements of a module M and show that the set s M of all strongly nilpotent elements of a uniserial module M defined over a commutative unital ring coincides with the classical prime radical cl M the intersection of all classical prime submodules of M.Definition 2.1 of [1] should state as follows: an element m of an R-module M is strongly nilpotent if m = 0 or for every sequence a 1 a 2 a 3 with a 1 = a, a n+1 ∈ a n Ra n for all n and 0 = am = a t m for some positive integer t, there exists a positive integer k ≥ t such that a k m = 0.

…”

“…Definition 2.1 of [1] should state as follows: an element m of an R-module M is strongly nilpotent if m = 0 or for every sequence a 1 a 2 a 3 with a 1 = a, a n+1 ∈ a n Ra n for all n and 0 = am = a t m for some positive integer t, there exists a positive integer k ≥ t such that a k m = 0.…”

“…The abstract of [1] should read as follows: We define nilpotent and strongly nilpotent elements of a module M and show that the set s M of all strongly nilpotent elements of a uniserial module M defined over a commutative unital ring coincides with the classical prime radical cl M the intersection of all classical prime submodules of M.…”