free modules造句

• Projective modules ( and thus free modules ) are always flat.
• In particular, P is a finitely generated free module.
• Stably-free modules vanish in reduced K-theory.
• Square matrix rings arise as endomorphism rings of free modules with finite rank.
• A finitely generated torsion-free module of a commutative PID is free.
• It is an isomorphism if " E " is a free module of finite rank.
• The concept and algorithms of Gr鯾ner bases have been generalized to submodules of free modules over a polynomial ring.
• A resolution is defined as " minimal " if the image in each module morphism of free modules
• Over a Noetherian integral domain, torsion-free modules are the modules whose only associated prime is 0.
• Given a modules, the free object on is the vector space or free module that has as a basis.
• This is equivalent to the statement that the polynomial ring is a free module over the ring of radical polynomials.
• Geometrically, finitely generated projective modules correspond to vector bundles over affine space, and free modules to trivial vector bundles.
• Ordinary matrices can be thought of as the coordinate representations of linear maps between vector spaces ( or free modules ).
• A free module is a projective module, but the converse may be wrong over some rings, such as Dedekind rings.
• This is equivalent to the statement that the ring of all polynomials is a free module over the ring of radical polynomials.
• Another issue that arises with non-finitely generated modules is that there are torsion-free modules which are not free.
• The relations to make the tensor product in the free module generated by the direct product can be done in similar ways.
• On the other hand, many important properties of free abelian groups may be generalized to free modules over a principal ideal domain.
• In algebra, a "'group ring "'is a free module and at the same time a group.
• In contrast, no choice is needed to prove that free modules are flat, so theorems about flat modules can still apply.