# Ideals of direct products of rings

## TL;DR

This paper investigates when ideals in direct products of rings can be decomposed into ideals of factors, providing necessary and sufficient conditions and characterizations for commutative rings and their varieties.

## Contribution

It extends known results from commutative unitary rings to general commutative rings, offering new criteria and characterizations for ideal decomposability in various varieties.

## Key findings

- Ideal decomposability does not always hold in general commutative rings.
- Necessary and sufficient conditions for ideal decomposability are established.
- A Mal'cev type condition characterizes decomposability in varieties of commutative rings.

## Abstract

It is known that an ideal of a direct product of commutative unitary rings is directly decomposable into ideals of the corresponding factors. We show that this does not hold in general for commutative rings and we find necessary and sufficient conditions for direct decomposability of ideals. For varieties of commutative rings we derive a Mal'cev type condition characterizing direct decomposability of ideals and we determine explicitly all varieties satisfying this condition.