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 ...
By convention, a ring has the multiplicative identity. But some authors do not require a ring to have the multiplicative identity; i.e., for them, a ring is a rng. For a rng R, a left ideal I is a subrng with the additional property that is in I for every and every . (Right and two-sided ideals are defined similarly.)
Therefore, by definition, any field is a commutative ring. The rational , real and complex numbers form fields. If R {\displaystyle R} is a given commutative ring, then the set of all polynomials in the variable X {\displaystyle X} whose coefficients are in R {\displaystyle R} forms the polynomial ring , denoted R [ X ] {\displaystyle R\left[X ...
The definition of a polynomial ring can be generalised by relaxing the requirement that the algebraic structure R be a field or a ring to the requirement that R only be a semifield or rig; the resulting polynomial structure/extension R[X] is a polynomial rig.
This extends the definition for commutative rings. 4. prime ring : A nonzero ring R is called a prime ring if for any two elements a and b of R with aRb = 0, we have either a = 0 or b = 0. This is equivalent to saying that the zero ideal is a prime ideal (in the noncommutative sense.) Every simple ring and every domain is a prime ring. primitive 1.
The maximal ring of quotients Q(R) (in the sense of Utumi and Lambek) of a Boolean ring R is a Boolean ring, since every partial endomorphism is idempotent. [6] Every prime ideal P in a Boolean ring R is maximal: the quotient ring R / P is an integral domain and also a Boolean ring, so it is isomorphic to the field F 2, which shows the ...
The Princess cut is the only true square diamond and one of the newest diamond shapes. With striking pointed corners, Princess cut diamonds have an undeniable presence that's perfect for ...
"Rings prevent collisions." The term "ring" only refers to the layout of the cables. It is true that there are no collisions on an IBM Token Ring, but this is because of the layer 2 Media Access Control method, not the physical topology (which again is a star, not a ring.) Token passing, not rings, prevent collisions. "Token passing happens on ...