Direct sums of ideals

AbstractWe survey various existence and uniqueness theorems for decompositions of finitely generated modules over commutative rings, as direct sums of ideals of the ring. These theorems generalize a theorem of Steinitz published in 1912. Much of the paper is expository. The main new result is the following uniqueness theorem (well known to be true for integral domains): Let Ai, Bi be ideals of the commutative ring R, and suppose that the R-modules A1 ⊕⋯⊕ Am and B1 ⊕⋯⊕ Bm are isomorphic. Then the ideal products A1⋯Am and B1⋯Bm are isomorphic as R-modules