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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.
The zero element of this ring coincides with the 0 of the Boolean algebra; the multiplicative identity element of the ring is the 1 of the Boolean algebra. This ring has the property that a · a = a for all a in A; rings with this property are called Boolean rings.
A completely distributive lattice L is called the free completely distributive lattice over a poset C if and only if there is an order embedding: such that for every completely distributive lattice M and monotonic function:, there is a unique complete homomorphism: satisfying =.
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]
The category of rings is a symmetric monoidal category with the tensor product of rings ⊗ Z as the monoidal product and the ring of integers Z as the unit object. It follows from the Eckmann–Hilton theorem , that a monoid in Ring is a commutative ring .
Division rings are the only rings over which every module is free: a ring R is a division ring if and only if every R-module is free. [9] The center of a division ring is commutative and therefore a field. [10] Every division ring is therefore a division algebra over its center. Division rings can be roughly classified according to whether or ...
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).