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A Boolean algebra can be interpreted either as a special kind of ring (a Boolean ring) or a special kind of distributive lattice (a Boolean lattice). Each interpretation is responsible for different distributive laws in the Boolean algebra. Similar structures without distributive laws are near-rings and near-fields instead of rings and division ...
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. Informally, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers.
Conversely, every distributive lattice is isomorphic to a ring of sets; in the case of finite distributive lattices, this is Birkhoff's representation theorem and the sets may be taken as the lower sets of a partially ordered set. [1] A family of sets closed under union and relative complement is also closed under symmetric difference and ...
The factor ring of a maximal ideal is a simple ring in general and is a field for commutative rings. [12] Minimal ideal: A nonzero ideal is called minimal if it contains no other nonzero ideal. Zero ideal: the ideal {}. [13] Unit ideal: the whole ring (being the ideal generated by ). [9]
Multiplication of inhomogeneous elements is defined by using the distributive property. A ring or module may be related to its associated graded ring or module through the initial form map . Let M be an R -module and I an ideal of R .
The importance of such polynomial rings relies on the high number of properties that they have in common with the ring of the integers. [ 2 ] Polynomial rings occur and are often fundamental in many parts of mathematics such as number theory , commutative algebra , and algebraic geometry .
Rings often do not support division. Thus, the commutator of two elements a and b of a ring (or any associative algebra) is defined differently by [,] =.The commutator is zero if and only if a and b commute.
An element x is called a dual distributive element if ∀y,z: x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z). In a distributive lattice, every element is of course both distributive and dual distributive. In a non-distributive lattice, there may be elements that are distributive, but not dual distributive (and vice versa).