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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 R-module
has a monic PI-cover,
where PI denotes the class of all
P-injective
right R-modules.
In the second part, for a nonempty subset X of a ring R, we introduce the notion of X-PP rings
which unifies PP rings,
PS rings and
nonsingular rings. Special attention is paid to J-PP rings,
where J is the
Jacobson radical of R.
It is shown that right J-PP 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.
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