  In this article, we investigate the avoidance property of ideals and rings.
Among the main results, a general version of the avoidance lemma is formulated.
It is shown that every idempotent ideal (and hence every pure ideal) has
avoidance. The avoidance property of arbitrary direct products of avoidance
rings is characterized. It is shown that every overring of an avoidance domain
is an avoidance domain. Next, we show that every avoidance $\mathbb{N}$-graded
ring whose base subring is a finite field is a PIR. It is also proved that the
avoidance property is preserved under flat ring epimorphisms. Dually, we
formulate a notion of strong avoidance, and show that it is reflected by pure
morphisms.
