# Categories of Algebraic Systems: Vector and Projective by M. Petrich

By M. Petrich

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The left right closure of I is trl and the right left closure is rtl. An ideal I is left right closed if trl = I and right left closed if rtl = I. Two ring theoretic questions immediately arise: which ideals are both left right and right left closed? 25], but only for special classes of rings. We do not deal with these general problems in this paper. Instead, we apply the ideal connection to the case that R = £, the endomorphism ring of a module M. Thus we have the lattice H of fully invariant submodules of M, the lattice X of ideals of £ and three Galois connections to keep track of.

Let (6): 0 -> H -^ An A G -> 0 be pure ezact with QE(A) = Q x Q. Suppose A « BI © Ci. T/ien: (a) ff is quasi-isomorphic to a completely decomposable qd group with the dual types of its rank one summands a subset of {dr(jBi),dr(C'i),dr(Si) V dr(Bi)} and (b) G is quasi-isomorphic to a completely decomposable qd group with the dual types of its rank one summands a subset of Proof. Let (5\) be the quasi-pure exact sequence obtained from (5) as described above. Apply the duality d to (<5i) to obtain a quasi-pure exact sequence of torsionfree groups: 0 -> dGi ^ dB^ @ dCf -^> dHi -> 0.

Define relations < on S by inclusion and ^ on F by G/H ^ G/K ifK