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In algebra, a free presentation of a module M over a commutative ring R is an exact sequence of R-modules:
Note the image under g of the standard basis generates M. In particular, if J is finite, then M is a finitely generated module. If I and J are finite sets, then the presentation is called a finite presentation; a module is called finitely presented if it admits a finite presentation.
Since f is a module homomorphism between free modules, it can be visualized as an matrix with entries in R and M as its cokernel.
A free presentation always exists: any module is a quotient of a free module: F → g M → 0 {\displaystyle F\ {\overset {g}{\to }}\ M\to 0} , but then the kernel of g is again a quotient of a free module: F ′ → f ker g → 0 {\displaystyle F'\ {\overset {f}{\to }}\ \ker g\to 0}. The combination of f and g is a free presentation of M. Now, one can obviously keep "resolving" the kernels in this fashion; the result is called a free resolution. Thus, a free presentation is the early part of the free resolution.