Let R be a ring (with 1) of zero
singular right ideal and let Q be its maximal right quotient ring; let 𝒩 be the class or
all (unitary) right R-modules of zero singular submodule. An element M of 𝒩 is said
to be an injective cogenerator for 𝒩 if M is an injective module and every element of
𝒩 can be embedded in a direct product of copies of M;M is said to be
a minimal injective cogenerator for 𝒩 if M is the only direct summand
of M, which is an injective cogenerator for 𝒩. This paper deals with the
question of existence and uniqueness of a minimal injective cogenerator for 𝒩
(and in 𝒩). If a minimal injective cogenerator for 𝒩 exists, then it is (a)
isomorphic to a minimal faithful direct summand of Q, (b) isomorphic to a direct
summand of every injective cogenerator for 𝒩 (and in 𝒩) and (c) unique (up to
isomorphism). Whether Q is (or is not) a prime ring, affects the structure,
though not the existence, of a minimal injective cogenerator for 𝒩: a minimal
injective cogenerator for 𝒩, if it exists, is (up to isomorphism) a faithful
minimal right ideal of Q iff Q is a prime ring and so in this case Q is a
minimal injective cogenerator 𝒩 iff Q is a division ring. On the other hand, if
RR is finite dimensional (Goldie) then a minimal injective cogenerator for
𝒩 exists; it is Q iff Q is (ring) isomorphic to a finite product of division
rings.
