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of an integral commutative ring $ A $ of the [[field of fractions]] of the ring $ A $

''of an associative ring'' ...plication, only a [[magma]]; it is called the multiplicative system of the ring.

...for fixed $g$ in a group; the corresponding maps in a [[monoid]], unital [[ring]] and other algebraic structures: see [[Conjugate elements]].

...ators. More precisely, a non-empty set $A$ is called a left polygon over a monoid $R$ if for any $\lambda \in R$ and $a \in A$ the product $\lambda a \in A$ ...quivalent to specifying a homomorphism $\phi$ from the monoid $R$ into the monoid of mappings of the set $A$ into itself that transforms 1 to the identity ma

...$, then $K(C)$ is the Witt–Grothendieck group of $k$ (cf. [[Witt ring|Witt ring]]). ...omorphism classes of vector bundles over a topological space $X$ (with the monoid addition induced by the direct sum) one again obtains the topological K-gro

''polynomial ring'' ...depends only on a finite number of variables. A ring of polynomials over a ring $R$ is a (commutative) [[free algebra]] with an identity over $R$; the set

A [[Noetherian ring|Noetherian ring]] $ R $ is called effective if its elements and ring operations can be described effectively as well as the problem of finding a

...hogonal sum form a monoid with cancellation; the canonical mapping of this monoid into its [[Grothendieck group|Grothendieck group]] is injective. The group the tensor product of forms induces on it the structure of a ring, which is known as the Witt–Grothendieck of $ k $[[#References|[7]]].

A transition from a commutative ring $ A $ to the ring of fractions (cf. [[Fractions, ring of|Fractions, ring of]]) $ A [ S ^ {-1} ] $,

Let $\Sigma$ be a finite alphabet. Let $\Sigma ^ { * }$ be the free [[Monoid|monoid]] generated by $\Sigma$. A subset $L \subseteq \Sigma ^ { * }$ is called a ...ut constants are considered here. However, that, or the fact that the free monoid is finitely generated, is not essential.

...f $\mathcal{C}$ is a [[Monoid|monoid]], then $Z \mathcal C $ is the monoid ring of $\mathcal{C}$ with coefficients in $Z$. The inclusion ${\cal C} \rightar ...{C}$ with coefficients in $M$, which is a [[Module|module]] over the group ring $Z \mathcal C $ (cf. also [[Cross product|Cross product]]). In this case th

...a monoid (a semi-group with a unit element) and an inner automorphism of a ring (associative with a unit element), which are introduced in a similar way us

...of two algebraic systems. The endomorphisms of any algebraic system form a monoid under the operation of composition of mappings, whose unit element is the i of a ring $ R $

...nally equivalent to the variety of all right modules over some associative ring if and only if the congruences on any algebra in $M$ commute, if finite fre

...e|Lie algebra, free]]) is freely generated (as a module over a commutative ring $ K $) ...lows: One has the direct sum decomposition (as a module over a commutative ring $ K $):

there are associated related structures: the monoid of all endomorphisms $ \mathop{\rm End} A $, ...nt lattices of subalgebras (simple universal algebras), with a commutative monoid of endomorphisms, with a $ 1 $-element group of automorphisms (rigid univ

A [[Monoid|monoid]] in the [[Category|category]] of all endomorphism functors on $ \mathfra 4) In the category of modules over a commutative ring $ R $,

is an associative algebra over a commutative ring $ K $ is the ring of integers, $ G $

...ngle object; conversely, every category consisting of a single object is a monoid. ...ras; this is based on the interpretation of the categoric definitions of a monoid and a comonoid in suitable categories of functors (see, for example, [[#Ref