Abstract algebra/Rings

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[edit] Introduction to Rings

Rings are algebraic structures designed to model and abstract the structure of the integers (), so that we can duplicate some of the processes in which integers are used, but in a more general setting. It will be helpful if you have familiarity with the concepts and theorems for groups, because we'll be using many of the same ideas and theorems.

Definition: A ring is a set R with two binary operations + and that satisfies the following properties:

For all

1. The ring forms an abelian group under the addition operation.
2. (R is closed under + )
3. ( + is associative)
4. (R contains an additive identitiy)
5. (R contains additive inverses)
6. ( + is commutative)
7. (R is closed under )
8. ( is associative)
9. and ( is distributive over + )

If you're already familiar with the concepts of groups and semigroups, we can compress the conditions above to:

1. is an abelian group
2. is a semigroup
3. is distributive over + .

We'll often use juxtaposition in place of , i.e., for .

[edit] Types of Rings

Definition: Note that a ring does not necessarily have the property of commutative multiplication. When it does, the ring is called a commutative ring.

Definition: Also, a ring does not necessarily have a multiplicative identity. A nonzero element of a ring that is an identity under multiplication is sometimes called a unity and is denoted 1.

Definition: Let R be a ring with a 1. An element is a unit and is invertible if there is an element such that rr ^{− 1} = 1. The set of all units is denoted by and is a group under the multiplication operation.

Definition: An element is a zero-divisor when there exists a nonzero such that rs = 0.

Observe that a zero-divisor may not be a unit.

Definition: A ring R with a is an integral domain when it is a commutative ring without non-zero zero divisors.

Definition: A ring R with a is a division ring or skew field if all non-zero elements are units i. e. under multiplication, it forms a group with its nonzero elements.

Definition: A field is a commutative division ring.

Theorem: If a ring has a unity, that unity is unique.

Proof: Let R be a ring with unity u. Now suppose x is another unity in R. Then , and .

Of course proving that a set with two operations satisfy all of the above conditions can be tedious. So, just as we did for groups, we note that if we're considering a subset of something that's already a ring, then our job is easier.

Definition: A subring S of a ring R is a subset of R that is a ring (under the same two operations as for R).

Theorem: If S is a subset of a ring R, then S is a subring of R if:

1. S is nonempty
2. S is closed under -
3. S is closed under

Examples:

1. The set of integers under addition and multiplication is a commutative ring with unity 1.
2. The set of even integers under addition and multiplication is a commutative ring without unity.
3. The set M₂ of square matrices composed of integers is a noncommutative ring with unity I_n.
4. The trivial subring is a subring of every ring.
5. The set of Gaussian integers is a subring of the complex numbers C.
6. Let R be a ring. Then let R[X] denote all the polynomials a₀Xⁿ + a₁X^{n − 1} + ... + a_{n − 1}X + a_n with coefficients in R. This polynomial is formally a sequence (a₀,a₁,a₂,...a_n) of elements in R with X as a placeholder. The product of two polynomials can be defined to be the polynomial where the coefficient of X^a is the convolution (a₀b_n + a₁b_{n − 1} + ... + a_nb₀). This ring is called the polynomial ring of the ring R.