direct sum of modules

A direct sum bolts modules together side by side into one bigger module, with no interaction between the pieces — like stacking independent number lines to build a plane. Each summand keeps its own arithmetic, and an element of the whole is just a tuple of elements, one from each piece. It is the way modules combine that loses no information: you can always read off each component.

For a family (M_i) of R-modules indexed by i in I, the direct sum ⊕ M_i is the set of tuples (m_i) with m_i in M_i and only finitely many m_i nonzero, under componentwise addition and scalar multiplication. The direct product ∏ M_i drops the finiteness condition; the two agree when I is finite. The direct sum is the coproduct in the category of R-modules: maps out of ⊕ M_i correspond to families of maps out of the M_i.

A module M is the internal direct sum of submodules N_1, ..., N_k, written M = N_1 ⊕ ... ⊕ N_k, if every element writes uniquely as n_1 + ... + n_k with n_i in N_i; equivalently M = N_1 + ... + N_k and each N_j meets the sum of the others only in 0. Decomposing a module into a direct sum of simpler (ideally indecomposable) summands is one of the central goals of structure theory.