In mathematics, especially in the field of module theory, the concept of pure submodule provides a generalization of direct summand, a type of particularly well-behaved piece of a module. Pure modules are complementary to flat modules and generalize Prüfer's notion of pure subgroups. While flat modules are those modules which leave short exact sequences exact after tensoring, a pure submodule defines a short exact sequence (known as a pure exact sequence) that remains exact after tensoring with any module. Similarly a flat module is a direct limit of projective modules, and a pure exact sequence is a direct limit of split exact sequences.
Let R be a ring (associative, with 1), let M be a (left) module over R, let P be a submodule of M and let i: P → M be the natural injective map. Then P is a pure submodule of M if, for any (right) R-module X, the natural induced map idX ⊗ i : X ⊗ P → X ⊗ M (where the tensor products are taken over R) is injective.
of (left) R-modules is pure exact if the sequence stays exact when tensored with any (right) R-module X. This is equivalent to saying that f(A) is a pure submodule of B.
Equivalent characterizationsedit
Purity of a submodule can also be expressed element-wise; it is really a statement about the solvability of certain systems of linear equations. Specifically, P is pure in M if and only if the following condition holds: for any m-by-nmatrix (aij) with entries in R, and any set y1, ..., ym of elements of P, if there exist elements x1, ..., xnin M such that
then there also exist elements x1′, ..., xn′ in P such that
C is a flat module if and only if the exact sequence is pure exact for every A and B. From this we can deduce that over a von Neumann regular ring, every submodule of everyR-module is pure. This is because every module over a von Neumann regular ring is flat. The converse is also true. (Lam & 1999, p.162) harv error: no target: CITEREFLam1999,_p.162 (help)
Suppose B is flat. Then the sequence is pure exact if and only if C is flat. From this one can deduce that pure submodules of flat modules are flat.
Suppose C is flat. Then B is flat if and only if A is flat.
If is pure-exact, and F is a finitely presentedR-module, then every homomorphism from F to C can be lifted to B, i.e. to every u : F → C there exists v : F → B such that gv=u.
Referencesedit
^For abelian groups, this is proved in Fuchs (2015, Ch. 5, Thm. 3.4)
Fuchs, László (2015), Abelian Groups, Springer Monographs in Mathematics, Springer, ISBN9783319194226
Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York: Springer-Verlag, ISBN978-0-387-98428-5, MR 1653294
December 15, 2023
pure, submodule, mathematics, especially, field, module, theory, concept, pure, submodule, provides, generalization, direct, summand, type, particularly, well, behaved, piece, module, pure, modules, complementary, flat, modules, generalize, prüfer, notion, pur. In mathematics especially in the field of module theory the concept of pure submodule provides a generalization of direct summand a type of particularly well behaved piece of a module Pure modules are complementary to flat modules and generalize Prufer s notion of pure subgroups While flat modules are those modules which leave short exact sequences exact after tensoring a pure submodule defines a short exact sequence known as a pure exact sequence that remains exact after tensoring with any module Similarly a flat module is a direct limit of projective modules and a pure exact sequence is a direct limit of split exact sequences Contents 1 Definition 2 Equivalent characterizations 3 Examples 4 Properties 5 ReferencesDefinition editLet R be a ring associative with 1 let M be a left module over R let P be a submodule of M and let i P M be the natural injective map Then P is a pure submodule of M if for any right R module X the natural induced map idX i X P X M where the tensor products are taken over R is injective Analogously a short exact sequence 0 A f B g C 0 displaystyle 0 longrightarrow A stackrel f longrightarrow B stackrel g longrightarrow C longrightarrow 0 nbsp of left R modules is pure exact if the sequence stays exact when tensored with any right R module X This is equivalent to saying that f A is a pure submodule of B Equivalent characterizations editPurity of a submodule can also be expressed element wise it is really a statement about the solvability of certain systems of linear equations Specifically P is pure in M if and only if the following condition holds for any m by n matrix aij with entries in R and any set y1 ym of elements of P if there exist elements x1 xn in M such that j 1 n a i j x j y i for i 1 m displaystyle sum j 1 n a ij x j y i qquad mbox for i 1 ldots m nbsp then there also exist elements x1 xn in P such that j 1 n a i j x j y i for i 1 m displaystyle sum j 1 n a ij x j y i qquad mbox for i 1 ldots m nbsp Another characterization is a sequence is pure exact if and only if it is the filtered colimit also known as direct limit of split exact sequences 0 A i B i C i 0 displaystyle 0 longrightarrow A i longrightarrow B i longrightarrow C i longrightarrow 0 nbsp 1 Examples editEvery direct summand of M is pure in M Consequently every subspace of a vector space over a field is pure Properties edit Lam amp 1999 p 154 harv error no target CITEREFLam1999 p 154 help Suppose 0 A f B g C 0 displaystyle 0 longrightarrow A stackrel f longrightarrow B stackrel g longrightarrow C longrightarrow 0 nbsp is a short exact sequence of R modules then C is a flat module if and only if the exact sequence is pure exact for every A and B From this we can deduce that over a von Neumann regular ring every submodule of every R module is pure This is because every module over a von Neumann regular ring is flat The converse is also true Lam amp 1999 p 162 harv error no target CITEREFLam1999 p 162 help Suppose B is flat Then the sequence is pure exact if and only if C is flat From this one can deduce that pure submodules of flat modules are flat Suppose C is flat Then B is flat if and only if A is flat If 0 A f B g C 0 displaystyle 0 longrightarrow A stackrel f longrightarrow B stackrel g longrightarrow C longrightarrow 0 nbsp is pure exact and F is a finitely presented R module then every homomorphism from F to C can be lifted to B i e to every u F C there exists v F B such that gv u References edit For abelian groups this is proved in Fuchs 2015 Ch 5 Thm 3 4 Fuchs Laszlo 2015 Abelian Groups Springer Monographs in Mathematics Springer ISBN 9783319194226Lam Tsit Yuen 1999 Lectures on modules and rings Graduate Texts in Mathematics No 189 Berlin New York Springer Verlag ISBN 978 0 387 98428 5 MR 1653294 Retrieved from https en wikipedia org w index php title Pure submodule amp oldid 1139453743, wikipedia, wiki, book, books, library,