Given a ring \(R\) and an ideal \(I\) of \(R\), a reduction of \(I\) is an ideal \(J\subseteq I\) such that \(JI^n=I^{n+1}\) for some integer \(n\geq 1\); the core of \(I\) is the intersection of all its reductions. Reductions are connected to the concept of integral closure of ideals, and have been deeply studied, mainly for Noetherian rings.
In this paper, the authors study reductions of ideals in the context of the so-called trivial ring extension (or Nagata idealization) of a ring by a module. Given a ring \(A\) and an \(A\)-module \(E\), the trivial ring extension of \(A\) by \(E\) is the ring \(R:=A\ltimes E\) (also denoted by \(A(+)E\)), whose additive group is the direct sum \(A\times E\) and whose multiplication is given by \((a,e)(a',e')=(aa',ae'+a'e)\).
The authors study this subject from two different points of view. In Section 2, they prove some general results about reductions and the core of ideals in the form \(I\ltimes F\), where \(I\) is an ideal of \(A\) and \(F\) a submodule of \(E\), relating them to the reductions and to the core of \(I\) inside \(A\). In Section 3, they prove results about general ideals of \(R=A\ltimes E\), but only under certain hypothesis on \(E\): they analyze the cases where \(E\) is a simple \(A\)-module, where \(E\) is a divisible \(A\)-module (and \(A\) is a domain), and where \(E\) is an \(A\)-module that is annihilated by a maximal ideal of \(A\). In the final Section 4, they construct some examples illustrating the results of the paper.