the lower left to the upper right of the diagram. The use of the exact sequence convention saves us from having to draw the zero morphisms from to and . If the sequence is right split (i.e., there is a morphism σ that maps to a -preimage of itself), then G is the semidirect product of the normal subgroup and the subgroup . If it is left split (i.e., there exists some such that ), then it must also be right split, and is a direct product decomposition of G. In general, the existence of a right split does not imply the existence of a left split; but in an abelian category (such as that of abelian groups), left splits and right splits are equivalent by the splitting lemma, and a right split is sufficient to produce a direct sum decomposition . In an abelian category, all monomorphisms are also normal, and the diagram may be extended by a second short exact sequence . In the second isomorphism theorem, the product SN is the join of S and N in the lattice of subgroups of G, while the intersection S ∩ N is the meet. The third isomorphism theorem is generalized by the nine lemma to abelian categories and more general maps between objects. Rings The statements of the theorems for rings are similar, with the notion of a normal subgroup replaced by the notion of an ideal. Theorem A (rings) Let R and S be rings, and let φ : R → S be a ring homomorphism. Then: The kernel of φ is an ideal of R, The image of φ is a subring of S, and The image of φ is isomorphic to the quotient ring R / ker(φ). In particular, if φ is surjective then S is isomorphic to R / ker(φ). Theorem B (rings) Let R be a ring. Let S be a subring of R, and let I be an ideal of R. Then: The sum S + I = {s + i | s ∈ S, i ∈ I } is a subring of R, The intersection S ∩ I is an ideal of S, and The quotient rings (S + I) / I and S / (S ∩ I) are isomorphic. Theorem C (rings) Let R be a ring, and I an ideal of R. Then If is a subring of such that