Search results
Results from the WOW.Com Content Network
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 ...
[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 ...
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 ...
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 ...
[1] [2] [3] These conditions played an important role in the development of the structure theory of commutative rings in the works of David Hilbert, Emmy Noether, and Emil Artin. The conditions themselves can be stated in an abstract form, so that they make sense for any partially ordered set. This point of view is useful in abstract algebraic ...
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 ...