Ring theory in the context of Field of sets

Julius Wilhelm Richard Dedekind (/ˈdeɪdɪkɪnd/; German: [ˈdeːdəˌkɪnt]; 6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His best known contribution is the definition of real numbers through the notion of Dedekind cut. He is also considered a pioneer in the development of modern set theory and of the philosophy of mathematics known as logicism.

Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures in general, not specific types of algebraic structures.For instance, rather than considering groups or rings as the object of study—this is the subject of group theory and ring theory— in universal algebra, the object of study is the possible types of algebraic structures and their relationships.

In mathematics, and more specifically in ring theory, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer (even or odd) results in an even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring in a way similar to how, in group theory, a normal subgroup can be used to construct a quotient group.

Among the integers, the ideals correspond one-for-one with the non-negative integers: in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number. However, in other rings, the ideals may not correspond directly to the ring elements, and certain properties of integers, when generalized to rings, attach more naturally to the ideals than to the elements of the ring. For instance, the prime ideals of a ring are analogous to prime numbers, and the Chinese remainder theorem can be generalized to ideals. There is a version of unique prime factorization for the ideals of a Dedekind domain (a type of ring important in number theory).

In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist a and b in the ring such that ab and ba are different. Equivalently, a noncommutative ring is a ring that is not a commutative ring.

Noncommutative algebra is the part of ring theory devoted to study of properties of the noncommutative rings, including the properties that apply also to commutative rings.

In ring theory, a branch of mathematics, the zero ring or trivial ring is the unique ring (up to isomorphism) consisting of one element. (Less commonly, the term "zero ring" is used to refer to any rng of square zero, i.e., a rng in which xy = 0 for all x and y. This article refers to the one-element ring.)

In the category of rings, the zero ring is the terminal object, whereas the ring of integers Z is the initial object.

In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.

In ring theory, a branch of mathematics, the radical of an ideal ${\displaystyle I}$ of a commutative ring is another ideal defined by the property that an element ${\displaystyle x}$ is in the radical if and only if some power of ${\displaystyle x}$ is in ${\displaystyle I}$. Taking the radical of an ideal is called radicalization. A radical ideal (or semiprime ideal) is an ideal that is equal to its radical. The radical of a primary ideal is a prime ideal.

This concept is generalized to non-commutative rings in the semiprime ring article.

In ring theory, a branch of mathematics, a ring R is a polynomial identity ring if there is, for some N > 0, an element P ≠ 0 of the free algebra, Z⟨X₁, X₂, ..., X_N⟩, over the ring of integers in N variables X₁, X₂, ..., X_N such that

for all N-tuples r₁, r₂, ..., r_N taken from R.

In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all proper ideals. In other words, I is a maximal ideal of a ring R if there are no other ideals contained between I and R.

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. The set of maximal ideals of a unital commutative ring R, typically equipped with the Zariski topology, is known as the maximal spectrum of R and is variously denoted m-Spec R, Specm R, MaxSpec R, or Spm R.