Maximal ideal - Citizendia

In mathematics, more specifically in ring theory, a maximal ideal is an ideal which is maximal (with respect to set inclusion) amongst all proper ideals, i. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, ring theory is the study of rings, Algebraic structures in which addition and multiplication are defined and have similar properties to those In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. e. which is not contained in any other proper ideal of the ring. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real

Maximal ideals are important because the quotient rings of maximal ideals are simple rings, and in the special case of unital commutative rings they are also fields. In Mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in Ring theory, quite similar to the In Abstract algebra, a simple ring is a non-zero ring that has no ideal besides the Zero ideal and itself In Mathematics, an algebra is unital (some authors say unitary) if it contains a multiplicative Identity element (or unit) i In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division Rings which contain only one maximal ideal are called local rings. In Mathematics, more particularly in Abstract algebra, local rings are certain rings that are comparatively simple and serve to describe what is called

Definition

Given a ring R and a proper ideal I of R (that is I ≠ R), I is called a maximal ideal of R if there exists no other proper ideal J of R so that I ⊂ J.

Examples

In the ring Z of integers the maximal ideals are the principal ideals generated by a prime number. In Ring theory, a branch of Abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single
More generally, this holds for any principal ideal domain.

Properties

In a commutative ring with unity, every maximal ideal is a prime ideal. In Mathematics, a prime ideal is a Subset of a ring which shares many important properties of a Prime number in the Ring of integers Maximal ideals can be directly characterized to be those ideals which are subsets of only two ideals: the improper ideal and the maximal ideal itself.

Krull's theorem (1929): Every commutative ring with a multiplicative identity has a maximal ideal. In Mathematics, more specifically in Ring theory, Krull's theorem, named after Wolfgang Krull, proves the existence of Maximal ideals in any

In a lattice diagram, maximal ideals are always directly joined to the biggest containing ring, as follows from the prime property. In the mathematical discipline known as Order theory, a Hasse diagram (ˈhɑːsə HAHS uh) named after Helmut Hasse (1898&ndash1979 is a

In a unital commutative ring, an ideal is maximal if and only if its factor ring is a field. In Mathematics, an algebra is unital (some authors say unitary) if it contains a multiplicative Identity element (or unit) i This fails in non-unital rings. For example, $4\mathbb{Z}$ is a maximal ideal in $2\mathbb{Z}$ , but $2\mathbb{Z}/4\mathbb{Z}$ is not a field.

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