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In a cochain complex of modules ...->C^(i-1)->^(d^(i-1))C^i->^(d^i)C^(i+1)->... the module Z^i of i-cocycles Z^i is the kernel of d^i, which is a submodule of C^i.

The cokernel of a group homomorphism f:A-->B of Abelian groups (modules, or abstract vector spaces) is the quotient group (quotient module or quotient space, respectively) ...

Two quantities are said to be different (or "unequal") if they are not equal. The term "different" also has a technical usage related to modules. Let a module M in an ...

For {M_i}_(i in I) a family of R-modules indexed by a directed set I, let sigma_(ij):M_i->M_j i<=j be an R-module homomorphism. Call (M_i,sigma_(ij)) a direct system over I ...

If, in the above commutative diagram of modules and module homomorphisms the columns and two upper rows are exact, then so is the bottom row.

A morphism f:Y->X in a category is an epimorphism if, for any two morphisms u,v:X->Z, uf=vf implies u=v. In the categories of sets, groups, modules, etc., an epimorphism is ...

The Cartesian product of a finite or infinite set of modules over a ring with only finitely many nonzero entries in each sequence.

A decomposition of a module into a direct sum of submodules. The index set for the collection of submodules is then called the grading set. Graded modules arise naturally in ...

In a chain complex of modules ...->C_(i+1)->^(d_(i+1))C_i->^(d_i)C_(i-1)->..., the module B_i of i-boundaries is the image of d_(i+1). It is a submodule of C_i and is ...

In a chain complex of modules ...->C_(i+1)->^(d_(i+1))C_i->^(d_i)C_(i-1)->... the module Z_i of i-cycles is the kernel of d_i, which is a submodule of C_i.