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2. Free algebra - Wikipedia

   en.wikipedia.org/wiki/Free_algebra

   For example, X 1 X 2 does not equal X 2 X 1. More generally, one can construct the free algebra R E on any set E of generators. Since rings may be regarded as Z-algebras, a free ring on E can be defined as the free algebra Z E . Over a field, the free algebra on n indeterminates can be constructed as the tensor algebra on an n-dimensional ...

3. Macaulay2 - Wikipedia

   en.wikipedia.org/wiki/Macaulay2

   Macaulay2 is built around fast implementations of algorithms useful for computation in commutative algebra and algebraic geometry. This core functionality includes arithmetic on rings, modules, and matrices, as well as algorithms for Gröbner bases, free resolutions, Hilbert series, determinants and Pfaffians, factoring, and similar.

4. Free object - Wikipedia

   en.wikipedia.org/wiki/Free_object

   In mathematics, the idea of a free object is one of the basic concepts of abstract algebra.Informally, a free object over a set A can be thought of as being a "generic" algebraic structure over A: the only equations that hold between elements of the free object are those that follow from the defining axioms of the algebraic structure.

5. Free Boolean algebra - Wikipedia

   en.wikipedia.org/wiki/Free_Boolean_algebra

   The Hasse diagram of the free Boolean algebra on two generators, p and q. Take p (left circle) to be "John is tall" and q (right circle)to be "Mary is rich". The atoms are the four elements in the row just above FALSE. The generators of a free Boolean algebra can represent independent propositions. Consider, for example, the propositions "John ...

6. List of algebras - Wikipedia

   en.wikipedia.org/wiki/List_of_algebras

   This is a list of possibly nonassociative algebras.An algebra is a module, wherein you can also multiply two module elements. (The multiplication in the module is compatible with multiplication-by-scalars from the base ring).

7. Free module - Wikipedia

   en.wikipedia.org/wiki/Free_module

   Every vector space is a free module, [1] but, if the ring of the coefficients is not a division ring (not a field in the commutative case), then there exist non-free modules. Given any set S and ring R, there is a free R-module with basis S, which is called the free module on S or module of formal R-linear combinations of the elements of S.