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In linear algebra, a linear relation, or simply relation, between elements of a vector space or a module is a linear equation that has these elements as a solution.
More precisely, if e 1 , … , e n {\displaystyle e_{1},\dots ,e_{n}} are elements of a module M over a ring R , a relation between e 1 , … , e n {\displaystyle e_{1},\dots ,e_{n}} is a sequence {\displaystyle } of elements of R such that
The relations between e 1 , … , e n {\displaystyle e_{1},\dots ,e_{n}} form a module. One is generally interested in the case where e 1 , … , e n {\displaystyle e_{1},\dots ,e_{n}} is a generating set of a finitely generated module M, in which case the module of the relations is often called a syzygy module of M. The syzygy module depends on the choice of a generating set, but it is unique up to the direct sum with a free module. That is, if S 1 {\displaystyle S_{1}} and S 2 {\displaystyle S_{2}} are syzygy modules corresponding to two generating sets of the same module, then they are stably isomorphic, which means that there exist two free modules L 1 {\displaystyle L_{1}} and L 2 {\displaystyle L_{2}} such that S 1 ⊕ L 1 {\displaystyle S_{1}\oplus L_{1}} and S 2 ⊕ L 2 {\displaystyle S_{2}\oplus L_{2}} are isomorphic.
Higher order syzygy modules are defined recursively: a first syzygy module of a module M is simply its syzygy module. For k > 1, a kth syzygy module of M is a syzygy module of a -th syzygy module. Hilbert's syzygy theorem states that, if R = K {\displaystyle R=K} is a polynomial ring in n indeterminates over a field, then every nth syzygy module is free. The case n = 0 is the fact that every finite dimensional vector space has a basis, and the case n = 1 is the fact that K is a principal ideal domain and that every submodule of a finitely generated free K module is also free.