MATH2401 Abstract Algebra (2): Ring

This article introduces unital rings and their ideals, then develops polynomial rings, quotients, adjoining constructions, product decompositions, fraction fields, and the prime and maximal ideal tests.

Collections of MATH2401 Abstract Algebra notes:

1. Group;
2. Ring (this article);
3. Module;
4. Field;
5. Action;
6. Galois Theory.

Throughout, rings have an identity , subrings contain the same identity, and ring homomorphisms preserve it. Ideals are two-sided unless specified.

Definitions

Definition. A ring is a set such that is an abelian group, multiplication is associative, and multiplication distributes over addition on both sides. A ring is commutative if multiplication commutes. A division ring is a nonzero ring in which every nonzero element is a unit; a commutative division ring is a field.

Definition. A subring contains and is closed under addition, additive inverses, and multiplication. A unit is for which for some ; the units form the group .

Definition. The characteristic of is the least positive integer such that , or if none exists. Here means a repeated sum, not ring multiplication by an element .

Polynomial rings

Let be commutative. The polynomial ring consists of the formal sums

with the usual coefficientwise addition and convolution product. For , is its largest nonzero exponent; set .

Definition. A zero divisor is a nonzero for which for some nonzero . An integral domain is a nonzero commutative ring with no zero divisors.

Proposition. is an integral domain if and only if is an integral domain.

Proof. If is a domain and , the leading coefficient of is the product of two nonzero elements, so . Conversely, embeds as the constant polynomials, so zero divisors in remain zero divisors in . ◻

Theorem. Division algorithm. Let , with and leading coefficient a unit. Then there are unique such that

In particular, this holds for every nonzero when is a field.

Proof. If or , take . Otherwise write , , , and , and set

Iteration gives with or . If also , then

If , then

a contradiction. Hence and . ◻

There is no ordering-free analogue for arbitrary multivariate division. Over a field, a chosen monomial order and an ordered list of divisors give the multivariate division algorithm; the remainder need not be unique unless additional hypotheses, such as a Gröbner basis, are imposed.

Homomorphisms and ideals

Definition. A ring homomorphism preserves , addition, and multiplication. Its kernel is . A bijective homomorphism is a ring isomorphism.

Definition. An ideal is an additive subgroup such that and for every . The ideal generated by is

For a commutative ring, is the principal ideal generated by .

Proposition. Kernels of ring homomorphisms are ideals. Conversely, every ideal is the kernel of the quotient homomorphism .

Proof. Additivity makes a kernel an additive subgroup, and for in the kernel. For the quotient map, exactly when . ◻

Proposition. There is a unique ring homomorphism sending to . Its kernel is , where .

Proof. The unique homomorphism is

 ◻

Theorem. Universal property of a polynomial ring. Let be commutative, let be a homomorphism, and let . There is a unique homomorphism extending and satisfying , namely

Taking gives the coefficientwise extension .

This substitution principle extends in the evident way to finitely many variables.

Proposition. A field has exactly the ideals and . Consequently, every unital homomorphism from a field to a nonzero ring is injective.

Proof. If an ideal contains , then , so . The kernel of a unital map from is not all of , hence it is zero. ◻

Theorem. If is a field, every ideal of is principal.

Proof. For a nonzero ideal , choose a monic of least degree. Division writes every as with ; since , minimality gives . Thus . The zero ideal is . ◻

Quotient rings

Theorem. Quotient ring construction. If , then is a ring under

The natural map , , is surjective and has kernel .

Theorem. First isomorphism theorem for rings. For a ring homomorphism ,

Proof. The map is well-defined, injective, and surjective onto , and it respects both ring operations and the identity. ◻

Theorem. Correspondence theorem for ideals. If is surjective, is an inclusion-preserving bijection between ideals containing and ideals of . Its inverse is .

Proof. The mutually inverse correspondences are

with

on the stated domains. ◻

Adjoining elements

Definition. If is a ring containing and , then is the smallest subring of containing and . When and are commutative,

Example. The coefficient ring is preserved. In

the coefficient map is injective because is monic. Thus contains an isomorphic copy of (and a root of ).

Example. The coefficient ring is destroyed. Let and . In one has ; multiplication by gives , and then gives . Hence is the zero ring and the coefficient map is not injective.

The evaluation map , , has image , hence . In an abstract construction , the coefficient map is injective exactly when ; a monic nonconstant is a useful sufficient condition.

Product rings and idempotents

The Cartesian product is a ring under componentwise operations.

Definition. An element is idempotent if . It is central if for every .

Theorem. Decomposition by an idempotent. If is a central idempotent of , then and are ideals and rings with identities and , and

For a commutative ring every idempotent is central.

Proof. Centrality makes both summands two-sided ideals. The displayed map is a ring homomorphism. Its inverse is : indeed and every decomposes as . ◻

Theorem. Chinese remainder theorem. If and , then

is surjective with kernel . Hence .

Proof. Suppose and choose , with . For every ,

because and . Thus the map is surjective. ◻

Fields of fractions

Let be an integral domain. On pairs with , put when . The class of is written .

Theorem. Field of fractions. These classes form a field containing an isomorphic copy of via .

Proof. Define and . Every nonzero has inverse , and is injective. ◻

Theorem. Universal property. If is an injective homomorphism to a field, there is a unique homomorphism extending , given by

Thus every field containing contains a unique -isomorphic copy of .

Proof. Any extension must send to the displayed value, proving uniqueness. Cross multiplication and injectivity of prove the formula is well-defined; direct calculation proves it is a homomorphism. ◻

Maximal and prime ideals

In this subsection is commutative.

Definition. A proper ideal is maximal if no proper ideal lies strictly between and . A proper ideal is prime if implies or .

Theorem. An ideal is maximal if and only if is a field.

Proof. By the correspondence theorem, ideals of correspond to ideals of containing . Thus is maximal exactly when the quotient has only and itself as ideals, which for a nonzero commutative ring is equivalent to every nonzero element being a unit. ◻

Theorem. An ideal is prime if and only if is an integral domain.

Proof. In , exactly when . The absence of nonzero zero divisors is therefore exactly the defining prime-ideal implication. ◻