nLab
principal ideal of a monoid
Contents
Context
Algebra
Monoid theory
monoid theory in algebra :
monoid , infinity-monoid
monoid object , monoid object in an (infinity,1)-category
Mon , CMon
monoid homomorphism
trivial monoid
submonoid , quotient monoid?
divisor , multiple? , quotient element?
inverse element , unit , irreducible element
ideal in a monoid
principal ideal in a monoid
commutative monoid
cancellative monoid
GCD monoid
unique factorization monoid
Bézout monoid
principal ideal monoid
group , abelian group
absorption monoid
free monoid , free commutative monoid
graphic monoid
monoid action
module over a monoid
localization of a monoid
group completion
endomorphism monoid
super commutative monoid
Contents
Definition
Given a monoid (or semigroup ) M M and an element a ∈ M a \in M , a left principal ideal in M M is a subset M a M a of M M such that for all m ∈ M m \in M , m a ∈ M a m a \in M a . Similarly, a right principal ideal in M M is a subset a M a M of M M such that for all m ∈ M m \in M , a m ∈ a M a m \in a M . Finally, a two-sided principal ideal , or simply principal ideal , in M M is a subset ⟨ a ⟩ \langle a \rangle that is both a left ideal and a right ideal.
Last revised on May 21, 2021 at 22:23:57.
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