Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Left inverse, right inverse, and invertible element of a monoid
Definition
Let be a monoid (Semigroup and monoid) and let . An element is
- a left inverse of when ;
- a right inverse of when ;
- a two-sided inverse, or simply an inverse, of when it is both, that is when .
The element is invertible, or a unit of , when it has a two-sided inverse. The set of units of is written .
The identity is always invertible, since exhibits as a two-sided inverse of itself.
Remarks
-
One-sided inverses are not enough on their own. The definition keeps the three notions apart because the sharp uniqueness statement, and hence the right to write , is about their interaction: in a monoid a left inverse of and a right inverse of are automatically equal (In a monoid, a left inverse and a right inverse of the same element are equal; hence an invertible element has exactly one inverse, and it is two-sided). Until that is proved, "the inverse of " is not legitimate language.
-
is a group under the restricted operation (The invertible elements of a monoid form a group under the restricted operation), and a monoid is a group exactly when (Group and abelian group).
Depends on
Used by
- 2ℤ is closed under addition, negation and multiplication and is not a subring of ℤ, because it does not contain 1 Counterexample
- A commutative monoid in which cancellation holds need not be a group: (ℕ, +) Counterexample
- Divisibility and associates in an integral domain Definition
- Division ring: a ring with 1 ≠ 0 in which every nonzero element is a unit Definition
- Group and abelian group Definition
- Invertible square matrices and similarity over a commutative ring Definition
- Irreducible and prime elements of an integral domain Definition
- Powers gⁿ: natural exponents in a monoid and integer exponents in a group, with g⁰ = e Definition
- Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides Definition
- Subgroup Definition
- The group μₙ(K) of n-th roots of unity in a field, and primitive n-th roots of unity Definition
- The product ring R × S with componentwise operations, its identity (1_R, 1_S) and its units R^× × S^× Definition
- The unit group (ℤ/n)^× and Euler's totient φ(n)=|(ℤ/n)^×| for n≥1 Definition
- Vector space over a field Definition
- (ℤ, +) is an abelian group, (ℤ, ·) is a commutative monoid that is not a group, and its group of units is {1, -1} Example
- For any field F, (F, +) and (F ∖ {0}, ·) are abelian groups; in particular (ℚ, +), (ℚ ∖ {0}, ·), (ℝ, +) and (ℝ ∖ {0}, ·) Example
- ℤ is an integral domain of characteristic 0 whose group of units is {1,-1}, so it is not a field: 2 is nonzero and not invertible Example
- ℤ sits inside ℚ as a subring that is not a subfield, so the inverse-closure clause of the subfield definition is doing work Example
- FALSE: In every commutative ring, each nonzero element is either a unit or a zero divisor False statement
- (ℤ, ·, 1) is a commutative monoid whose group of units is {1, -1}; equivalently u ∣ 1 holds exactly for u = 1 and u = -1 Lemma
- A ring homomorphism satisfies f(0) = 0, f(-a) = -f(a) and f(ma) = m f(a) for m ∈ ℤ, carries units to units, and has a subring as its image; composites of ring homomorphisms are ring homomorphisms Lemma
- Every commutative division ring is a field, so "field" and "commutative division ring" name the same structures and the published definition and the ring-theoretic one agree Lemma
- Every field is a commutative ring with 1 ≠ 0; it is an integral domain, and it is a commutative division ring Lemma
- Every prime element of an integral domain is irreducible Lemma
- In a group e⁻¹ = e, (g⁻¹)⁻¹ = g and (gh)⁻¹ = h⁻¹g⁻¹, the order of the last product being essential Lemma
- In a monoid, a left inverse and a right inverse of the same element are equal; hence an invertible element has exactly one inverse, and it is two-sided Lemma
- Sym(X) is a group under composition, and it is non-abelian whenever X has at least three distinct elements Lemma
- The invertible elements of a monoid form a group under the restricted operation Lemma
- The units of a ring are the invertible elements of its multiplicative monoid, and R^× is a group under multiplication; 0 ∈ R^× only in the zero ring Lemma
- An incidence function is convolution-invertible if and only if every diagonal value is a unit Theorem
- Equivalence classes of degree-one representations are exactly homomorphisms G→ k^×; equivalently they factor through G/G', and they form an abelian group Theorem
- ℍ is a division ring that is not commutative, hence not a field: q⁻¹ = q̄ / N(q) for q ≠ 0, while ij = k and ji = -k Theorem
Dependency tree · two levels
6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Inverse element (Wikipedia) (standard reference, not scraped)
- Unit (ring theory) (Wikipedia) (standard reference, not scraped)