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Given a module A and submodules M, N we may consider the intersection and sum of M and N. In an abelian category, given two monics f:M->A and g:N->A, we can form the intersection by taking the pullback of f,g, and the pullback object is a subobject of M,N because pullbacks of monics are monics. The best way I can think of taking the sum M+N is to factor f,g through the coproduct of M,N, then take the image of this arrow, but this doesn't use the fact that f,g are monic (unlike for intersection). Is there any better way of expressing M+N?

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Take the pushout of one of the pullback "projections" $i: M \cap N \hookrightarrow M$ along the other $j: M \cap N \hookrightarrow N$. This gives the two inclusions $M \to M + N$, $N \to M + N$, i.e., the pushout is the join $M + N$ in the lattice of subobjects of $A$.

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