abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...

, a

module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Mo ...

''M'' over a

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

''R'' is called torsionless if it can be embedded into some

direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...

''R''^''I''. Equivalently, ''M'' is torsionless if each non-zero element of ''M'' has non-zero image under some ''R''-linear functional ''f'': :

f\in M^=\operatorname_R(M,R),\quad f(m)\ne 0.

This notion was introduced by

Hyman Bass Hyman Bass (; born October 5, 1932)

injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contraposi ...

. If this map is bijective then the module is called reflexive. For this reason, torsionless modules are also known as semi-reflexive. * A unital

free module In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a fie ...

is torsionless. More generally, a

direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mo ...

of torsionless modules is torsionless. * A free module is reflexive if it is finitely generated, but for some rings there are also infinitely generated free modules that are reflexive. For instance, the direct sum of countably many copies of the integers is a reflexive module over the integers, see for instance. * A submodule of a torsionless module is torsionless. In particular, any

projective module In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of free modules. Various equivalent characterizati ...

over ''R'' is torsionless; any left ideal of ''R'' is a torsionless left module, and similarly for the right ideals. * Any torsionless module over a

domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function *Do ...

is a

torsion-free module In algebra, a torsion-free module is a module over a ring such that zero is the only element annihilated by a regular element (non zero-divisor) of the ring. In other words, a module is ''torsion free'' if its torsion submodule is reduced to ...

, but the converse is not true, as Q is a torsion-free Z-module which is ''not'' torsionless. * If ''R'' is a commutative ring which is an

integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...

and ''M'' is a finitely generated torsion-free module then ''M'' can be embedded into ''R''^''n'' and hence ''M'' is torsionless. * Suppose that ''N'' is a right ''R''-module, then its dual ''N''^∗ has a structure of a left ''R''-module. It turns out that any left ''R''-module arising in this way is torsionless (similarly, any right ''R''-module that is a dual of a left ''R''-module is torsionless). * Over a Dedekind domain, a finitely generated module is reflexive if and only if it is torsion-free. * Let ''R'' be a Noetherian ring and ''M'' a reflexive finitely generated module over ''R''. Then

M \otimes_R S

is a reflexive module over ''S'' whenever ''S'' is flat over ''R''.

Relation with semihereditary rings

Stephen Chase proved the following characterization of

semihereditary ring In mathematics, especially in the area of abstract algebra known as module theory, a ring ''R'' is called hereditary if all submodules of projective modules over ''R'' are again projective. If this is required only for finitely generated submodul ...

s in connection with torsionless modules: For any ring ''R'', the following conditions are equivalent: * ''R'' is left semihereditary. * All torsionless right ''R''-modules are flat. * The ring ''R'' is left coherent and satisfies any of the four conditions that are known to be equivalent: ** All right ideals of ''R'' are flat. ** All left ideals of ''R'' are flat. ** Submodules of all right flat ''R''-modules are flat. ** Submodules of all left flat ''R''-modules are flat. (The mixture of left/right adjectives in the statement is ''not'' a mistake.)

References

*Chapter VII of *{{Citation , last1=Lam , first1=Tsit-Yuen , title=Lectures on modules and rings , publisher=

Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 ...

, location=Berlin, New York , series=Graduate Texts in Mathematics No. 189 , isbn=978-0-387-98428-5 , mr=1653294 , year=1999 Module theory

Relation with semihereditary rings

See also

References