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S can be equipped with operations making it a ring such that the inclusion map S → R is a ring homomorphism. For example, the ring of integers is a subring of the field of real numbers and also a subring of the ring of polynomials [] (in both cases, contains 1, which is the multiplicative identity of the larger rings).
To simplify the description all rings are assumed to be commutative. The non-commutative case is discussed in detail in the respective articles. Ideals are important because they appear as kernels of ring homomorphisms and allow one to define factor rings. Different types of ideals are studied because they can be used to construct different ...
Maximal ideals are important because the quotients of rings by maximal ideals are simple rings, and in the special case of unital commutative rings they are also fields. In noncommutative ring theory, a maximal right ideal is defined analogously as being a maximal element in the poset of proper right ideals, and similarly, a maximal left ideal ...
In mathematics, specifically ring theory, a principal ideal is an ideal in a ring that is generated by a single element of through multiplication by every element of . The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset generated by a single element , which is to say the set of all elements less than or equal to in .
The principal rings constructed in Example 5 above are always Artinian rings; in particular they are isomorphic to a finite direct product of principal Artinian local rings. A local Artinian principal ring is called a special principal ring and has an extremely simple ideal structure: there are only finitely many ideals, each of which is a ...
These elements already have images but they are not the highest quality. These elements aren't too rare, so it shouldn't be too hard to find other pictures. Google's probably the best bet here but it may also be worth asking one of Wikipedia's element photographers (a certain person comes to mind) to see if they can get a better photo. Barium ...
An element p of a commutative ring R is said to be prime if it is not the zero element or a unit and whenever p divides ab for some a and b in R, then p divides a or p divides b.With this definition, Euclid's lemma is the assertion that prime numbers are prime elements in the ring of integers.
As an example, the nilradical of a ring, the set of all nilpotent elements, is not necessarily an ideal unless the ring is commutative. Specifically, the set of all nilpotent elements in the ring of all n × n matrices over a division ring never forms an ideal, irrespective of the division ring chosen. There are, however, analogues of the ...