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A ring is a set R equipped with two binary operations [a] + (addition) and ⋅ (multiplication) satisfying the following three sets of axioms, called the ring axioms: [1] [2] [3] R is an abelian group under addition, meaning that: (a + b) + c = a + (b + c) for all a, b, c in R (that is, + is associative). a + b = b + a for all a, b in R (that ...
A standard 8 + 1 ⁄ 2 by 11 inches (220 mm × 280 mm) sheet of paper has three holes with spacing of 4 + 1 ⁄ 4 inches (110 mm). There is a variant for half-letter size pages (8 + 1 ⁄ 2 by 5 + 1 ⁄ 2 inches or 220 mm × 140 mm), whose three rings are 2 + 3 ⁄ 4 inches (70 mm) apart. "Ledger" size binders hold 11-by-17-inch (28 by 43 cm ...
[5] A right ideal is defined similarly, with the condition replaced by . A two-sided ideal is a left ideal that is also a right ideal. If the ring is commutative, the three definitions are the same, and one talks simply of an ideal. In the non-commutative case, "ideal" is often used instead of "two-sided ideal".
5.1 The ring of integers of a number field. 5.2 The coordinate ring of an algebraic variety. 5.3 Ring of invariants. 6 History. 7 Notes. 8 References. Toggle the ...
Consider the two maps g 1 and g 2 from Z[x] to R that map x to r 1 and r 2, respectively; f ∘ g 1 and f ∘ g 2 are identical, but since f is a monomorphism this is impossible. However, surjective ring homomorphisms are vastly different from epimorphisms in the category of rings. For example, the inclusion Z ⊆ Q is a ring epimorphism, but ...
One defines the ring of integers of a non-archimedean local field F as the set of all elements of F with absolute value ≤ 1; this is a ring because of the strong triangle inequality. [12] If F is the completion of an algebraic number field, its ring of integers is the completion of the latter's ring of integers. The ring of integers of an ...
One basis of M 2 (C) consists of the four matrix units (matrices with one 1 and all other entries 0); another basis is given by the identity matrix and the three Pauli matrices. A matrix ring over a field is a Frobenius algebra, with Frobenius form given by the trace of the product: σ(A, B) = tr(AB).
Not every unital associative algebra over F 2 is a Boolean ring: consider for instance the polynomial ring F 2 [X]. The quotient ring R / I of any Boolean ring R modulo any ideal I is again a Boolean ring. Likewise, any subring of a Boolean ring is a Boolean ring. Any localization RS −1 of a Boolean ring R by a set S ⊆ R is a Boolean ring ...