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| 1 | Nilpotent |
| nilpotent elements from a commutative ring form an ideal; this is a consequence of the binomial theorem. This ideal is the nilradical of the ring. Every nilpotent element in a commutative ring is...In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer ... | |
| 2 | List of commutative algebra topics |
| This is a list of commutative algebra topics , by Wikipedia page. commutative ring ring ideal, maximal ideal, prime ideal ring monomorphism, ring epimorphism, ring isomorphism integral domain quotient field product of rings Boolean algebra zero divisor nilpotent, reduced (ring theory) annihilator ... | |
| 3 | Radical of an ideal |
| . First we will show that the nilpotent elements of R form an ideal N . Let a and b be nilpotent elements of R with a n = 0 and b m = 0. We will show that a+b is nilpotent. We can use the binomial... expression vanish. Thus a+b is nilpotent, and hence in N . To finish checking that N is an ideal, we ... | |
| 4 | Glossary of ring theory |
| of maximal ideals . Nilradical : The set of all nilpotent elements in a commutative ring forms an..." decomposition. A semisimple ring is also Noetherian , and has no nilpotent ideals. A ring can be... divides a or x divides b . Nilpotent : An element r of R is nilpotent if there exists a positive ... | |
| 5 | Jordan normal form |
| a sum M' + M* where M' is diagonalizable, M* is nilpotent, and M' commutes with M*. The way the... + N, where N is the special nilpotent matrix with (i,j)th entry 1 if i = j+1, and otherwise 0 (acts... the structure theorem for finitely-generated modules over principal ideal domains, of which it is a ... | |
| 6 | Jacobson radical |
| ) is a nilpotent ideal. Note however that in general the Jacobson radical need not contain every... In ring theory, a branch of abstract algebra, the Jacobson radical of a ring R is an ideal of R... can be defined in the following equivalent ways: the intersection of all maximal left ideals. the ... | |
| 7 | Algebraic variety |
| polynomial ring over K by a prime ideal. A projective algebraic variety was the closure in projective space... a big step technically. More serious is to allow nilpotents in the sheaf of rings. A nilpotent in a... the categorical point of view, nilpotents must be allowed, in order to have finite limits of ... | |
| 8 | Maximal ideal |
| In mathematics the term ideal refers to two different concepts: either to ideals in ring theory or to ideals in order theory. However, in both cases ideals are certain sets which can be partially ordered via subset inclusion. Consequently, a maximal ideal is a maximal element in the corresponding ... | |
| 9 | Ideal (mathematics) |
| In mathematics, the term ideal has multiple meanings. ideals are special subsets of a ring considered in abstract algebra. ideals appear in order theory and are a special kind of lower sets of an order. In physics, ideal may refer to The ideal gas law which governs the pressure of an ideal gas ... | |
| 10 | Ideality |
| Ideality is a faculty from the discipline of Phrenology. Definition Ideality describes the disposition towards perfection, towards beauty and refinement in all aspects of life. Localisation Ideality is located on the temples, above and behind Constructiveness . Ideality creates the width of the ... | |
| 11 | List of abstract algebra topics |
| Classification of finite simple groups Composition series Solvable group Nilpotent group... ideal Principal ideal Maximal ideal, prime ideal Jacobson radical Radical of an ideal Simple ring Product of rings Unit (ring theory) Idempotent Nilpotent, reduced (ring theory) Zero divisor Integral ... | |
| 12 | Primitive ideal |
| A left primitive ideal is the annihilator of a simple left module. A right primitive ideal is defined similarly. Note that (despite the name) left and right primitive ideals are always two-sided ideals. The quotient of a ring by a primitive ideal is a primitive ring ... | |
| 13 | Lie algebra |
| Lie algebras, see the Lie group article. Homomorphisms, Subalgebras and Ideals A homomorphism φ : g... subalgebra is then itself a Lie algebra. An ideal of the Lie algebra g is a subspace h of g such that [ a, y] ∈ h for all a ∈ g and y ∈ h . All ideals are subalgebras. If h is an ideal of g ... | |
| 14 | Quadratic field |
| factorization into ideals Any prime number p gives rise to an ideal p .O K in the ring of integers O K of a quadratic field K . In line with general theory, this may be a prime ideal, or a product of two distinct prime ideals of O K, or the square of a prime ideal of O K. The third case happens ... | |
| 15 | Ideal solution |
| In chemistry, an ideal solution is a solution where the enthalpy of solution is zero. The closer to zero the enthalpy of solution is, the more "ideally" the solution behaves. This becomes important in properties such as colligative properties, where the calculated values hold truer the more ideal ... |