algebra, a monoid ring is a ring constructed from a ring and a monoid, just as a group ring is constructed from a ring and a group. Let R be a ring and let...

naturally grades the corresponding group ring; similarly, monoid rings are graded by the corresponding monoid. An (associative) superalgebra is another...

is a free monoid. Transition monoids and syntactic monoids are used in describing finite-state machines. Trace monoids and history monoids provide a foundation...

In mathematics, the plactic monoid is the monoid of all words in the alphabet of positive integers modulo Knuth equivalence. Its elements can be identified...

noncommuting variables with coefficients in the ring R is the monoid ring R[N], where the monoid N is the free monoid on n letters, also known as the set of all...

the analogous case of groups) it may be called an abelian semigroup. A monoid is an algebraic structure intermediate between semigroups and groups, and...

noncommutative rings are given by rings of square matrices or more generally by rings of endomorphisms of abelian groups or modules, and by monoid rings. Representation...

In category theory, a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) (M, μ, η) in a monoidal category (C, ⊗, I) is...

rigs are rings without negative elements. (Akin to using rng to mean a ring without a multiplicative identity.) The term dioid (for "double monoid") has...

Stanley–Reisner ring was used to characterize the numbers of faces in each dimension of simplicial polytopes. Every ring can be thought of as a monoid in Ab, the...

algebra of a monoid is a generalization of the monoid ring that allows for infinite sums of elements of a ring. Suppose that S is a monoid with the property...

groups) or over Mon (the category of monoids). Specifically, there are forgetful functors A : Ring → Ab M : Ring → Mon which "forget" multiplication and...

Coxeter groups – then the group ring becomes a filtered algebra. Group algebra of a locally compact group Monoid ring Kaplansky's conjectures Group representation...

rings Quotient ring Matrix ring Endomorphism ring Polynomial ring Formal power series Monoid ring, Group ring Localization of a ring Tensor algebra Symmetric...

right adjoint to F. From monoids and groups to rings. The integral monoid ring construction gives a functor from monoids to rings. This functor is left adjoint...

consisting of square matrices with entries in formal variables. monoid A monoid ring. Morita Two rings are said to be Morita equivalent if the category of modules...

X_{2}X_{3}X_{1}^{4}X_{4}} . The non-commutative polynomial ring may be identified with the monoid ring over R of the free monoid of all finite words in the Xi. Since the...

can linearly extend the monoid homomorphisms  fi : M → k to k-algebra homomorphisms Fi : k[M] → k, where k[M] is the monoid ring of M over k. Then, by linearity...

follows that the set of all endomorphisms of X forms a monoid, the full transformation monoid, and denoted End(X) (or EndC(X) to emphasize the category...

algebra that is also a commutative ring. The definition is equivalent to saying that a unital associative R-algebra is a monoid object in R-Mod (the monoidal...

In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals. If the chain condition is satisfied...

integral domain is a ring for which the set of nonzero elements is a commutative monoid under multiplication (because a monoid must be closed under multiplication)...

generalized to any monoid object ⁠ ( R , ⊗ ) {\displaystyle (R,\otimes )} ⁠, where R {\displaystyle R} is the object where the monoid structure has been...

the category of algebras over commutative ring R, take R[N] → R[Z], where R[G] is the monoid ring of the monoid G and the morphism is induced by the inclusion...

In abstract algebra, a branch of mathematics, an affine monoid is a commutative monoid that is finitely generated, and is isomorphic to a submonoid of...

commutative monoids. Therefore, is it possible to define A ∞ {\displaystyle A_{\infty }} -ring spectra and E ∞ {\displaystyle E_{\infty }} -ring spectra as...

symbols x and y, then the monoid ring F [ G ] {\displaystyle F[G]\,} does not satisfy any Ore condition, but it is a free ideal ring and thus indeed a subring...

associativity is not assumed (but not excluded, either). Given an integer n, the ring of real square matrices of order n is an example of an associative algebra...

category may also be viewed as a "categorification" of an underlying monoid, namely the monoid whose elements are the isomorphism classes of the category's objects...

In algebra, a division ring, also called a skew field (or, occasionally, a sfield), is a nontrivial ring in which division by nonzero elements is defined...