Rings

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RINGS

Some definitions:

Ring. A set S with two binary operations, called addition and multiplication, which have the following properties:

1. Closed under addition.

a,b S implies a + b S

2. Associative Law holds under addition.

a + (b + c) = (a + b) + c for any a,b,c in S

3. Additive Identity 0 exists under addition.

a + 0 = 0 + a = a for all a in S

4. Additive Inverse exists for every element under addition.

a + (-a) = (-a ) + a = 0 for all a in S

5. Commutative Law holds under addition.

a + b = b + a for all a,b in S

6. Closed under multiplication.

a,b S implies ab S

7. Associative Law holds under multiplication.

a(bc) = (ab)c for any a,b,c in S

8. Left Distributive Law holds -- multiplication over addition

a(b+c) = ab + ac for any a,b,c in S

9. Right Distributive Law holds -- multiplication over addition

(a+b)c = ac + bc for any a,b,c in S

A ring need have no multiplicative identity elements or multiplicative inverses. It may have one or more multiplicative identity elements. It may have one or more right multiplicative identity elements but no left identity element and vice versa. If it has a left and right multiplicative identity element simultaneously, they are equal.

Commutative ring. A ring in which multiplication is commutative.

Ring with unit element (or ring with unity). A ring which has an identity element under multiplication.

Division ring. A ring whose nonzero elements form a group under multiplication.

Syn. Skew field

A division ring has no zero divisors (i.e. there are no two nonzero elements a and b for which ab = 0).

A commutative division ring is a field.

Examples of rings.

1. The set of all nxn matrices under the operations of matrix addition and matrix multiplication are non-commutative rings.

2. The set of all even integers is a commutative ring.

3. The set of integers modulo n is a commutative ring with unit element.

4. Sets of polynomials in given variables x₁, x₂, ... ,x_n with coefficients from any fixed ring or field under the usual operations of addition and multiplication.

5. The set of all continuous functions on a certain domain under the usual operations of addition and multiplication.

6. The set of linear transformations of a linear space or a Hilbert space.

Complex. A complex is any non-empty subset H of a ring R.

Operations of Complexes.

Addition. The sum H + K of two complexes H and K is the complex consisting of all sums

h + k for all h H and k K.

Multiplication. The product of two complexes H and K is the complex consisting of all products hk where h H and k K.

Subring. A subset of a ring R which is itself a ring with respect to the operations of R. Because any ring is a group with respect to the operation of addition, a subring of a ring R is a subgroup of the additive group of R.

Conditions for a subset of a ring to be a subring

● A subset S of a ring R is a subring of R if and only if S is non-empty and a - b and ab are in S for all a,b S.

● A subset S of a ring R is a subring of R if and only if S is non-empty and

(a) S is closed with respect to the ring operations

(b) for each a S, -a S

Because rings are, by definition, Abelian with regard to addition, any subring of a ring is necessarily normal with regard to addition.

Proper and Improper subrings. Among the subrings of a ring R are the subrings {0} and R itself. The rings {0} and R are called improper. Other subrings, if any, are called proper.

Divisors of zero. If ab = 0 and a ≠ 0 and b ≠ 0, then a and b are called divisors of zero. a is a left zero-divisor and b is a right zero-divisor.

Left Cancellation Law. If c ≠ 0 and ca = cb, then a = b.

Right Cancellation Law. If c ≠ 0 and ac = bc, then a = b.

Subtraction. The additive inverse of an element a is denoted as -a and called negative a. Subtraction is thus possible and unique in a ring R.

Theorems on rings.

1] Let R be a ring and let “0" be the identity element under addition. Then for every a in R:

1) a0 = 0

2) 0a = 0

2] For every a,b in a ring R:

1) -a = (-1)a

2) -(-a) = a

3) -(a+b) = (-a) + (-b)

4) (-a)(-b) = ab

5) a(-b) = (-a)b = -(ab) = -ab

6) a(b-c) = ab - ac

3] For every a,b in a ring R and every m,n in I⁺ (the set of positive integers):

1) a^maⁿ = a^{m+ n}

2) (a^m)ⁿ = a^mn

and if R is commutative

3) (ab)ⁿ = aⁿbⁿ

4] For every a,b in a ring R and every m,n I (set of all integers):

1) n(a + b) = na + nb

2) (m + n)a = ma + nb

3) n(ab) = (na)b = a(nb)

5] In a ring the right cancellation law for multiplication is equivalent to the statement that there are no divisors of zero.

6] In a ring the left cancellation law for multiplication is equivalent to the statement that there are no divisors of zero.

7] In a ring the left and right cancellation laws hold for addition.

Isomorphisms on rings. An isomorphism between two rings R and R* is a one-to-one correspondence between their elements that preserves ring operations. Specifically it is a correspondence such that if elements x and y of R correspond to elements x* and y* of R* then the xy must correspond to x*y* and x + y to x* + y*

Alternate definition. An isomorphism between two rings R and R* is a one-to-one mapping

f R → R* such that f(xy) = f(x)f(y) and f(x + y) = f(x) + f(y) for all x,y R.

Homomorphisms on rings A homomorphism between two rings R and R* is a mapping from ring R to ring R* that preserves ring operations. Specifically it is a mapping from R onto R* such that if elements x and y of R correspond to x* and y* of R*, then xy must correspond to x*y* and x + y to x* + y*.

Alternate definition. A homomorphism between two rings R and R* is a one-to-one mapping

f :R → R* such that f(xy) = f(x)f(y) and f(x + y) = f(x) + f(y) for all x,y. R.

Integral domains and fields are rings and the above definitions for isomorphisms and homomorphisms also apply to integral domains and fields.

References.

A.D. Aleksandrov, A.N. Kolmogorov, M.A. Lavrentev, editors. Mathematics, its Content, Methods and Meaning. Volume III.

James / James. Mathematics Dictionary.

Frank Ayres. Modern Algebra. (Schaum)

Joong Fang. Abstract Algebra. (Schaum)

Garrett Birkhoff, Saunders Mac Lane. A Survey of Modern Algebra.

R. A. Beaumont, R. W. Ball. Introduction to Modern Algebra and Matrix Theory.

B. L. Van der Waerden. Modern Algebra.

Dinkins. Abstract Mathematical Systems.