Abstract. This
paper consists of two parts. In the first part, it is proven that a ring R is right PP if and only if every
right Rmodule
has a monic PIcover,
where PI denotes the class of all
Pinjective
right Rmodules.
In the second part, for a nonempty subset X of a ring R, we introduce the notion of XPP rings
which unifies PP rings,
PS rings and
nonsingular rings. Special attention is paid to JPP rings,
where J is the
Jacobson radical of R.
It is shown that right JPP rings lie strictly
between right PP rings
and right PS rings.
Some new characterizations of (von Neumann) regular rings and semisimple
Artinian rings are also given.
