12. Rings 1 Rings Many of the groups with which we are familiar are arithmetical in nature, and they tend to share key structures that combine more than one operation. Example: Z, Q, R, and C are an Abelian groups under addition; subtraction is understood as a form of addition ( a b = a + ( b)) rather than as an independent operation in these groups. Multiplication provides another associative (and commutative) operation: each of these sets is closed under multiplication, with identity element 1, but none of them is a group under multiplication because closure under the taking of multiplicative inverses fails. In fact, 0 has no multiplicative inverse of any sort, so 0, the additive identity, will have special multiplicative properties not shared by other integers. Moreover, these two operations, addition and multiplication, are linked by virtue of the distributive property: a(b + c) = ab + ac; multiplication distributes over addition. Note that Z, Q, R, and C are nearly groups under multiplication: they are closed under this associative operation, and 1 behaves as an identity. But not all elements have inverses: indeed, 0 is unique in that it fails to have a multiplicative
12. Rings 2 inverse in any of these groups. We formalize these characteristics by defining a semigroup to be any set closed under an associative binary operation. (In particular, any group is also a semigroup.) But if we remove 0 from each of Z, Q, R, and C, we obtain the respective subsets Z*, Q*, R*, and C*; in the last three of these four, every element has a multiplicative inverse. The properties we observe here lead to the following definitions. A set R is called a ring if it is an Abelian group under (an operation which we conventionally signify as) addition, it is a semigroup under (an operation which we conventionally signify as) multiplication, and multiplication distributes over addition. If R contains an element that acts as a multiplicative identity, we call it a ring with unity. (Others call this a unitary ring. Still others require rings to contain a multiplicative identity by definition and call a ring without such an element a ring without unity, or a rng.) R is a commutative ring if its multiplication is always commutative, and a noncommutative ring otherwise.
12. Rings 3 Examples: Naturally, Z, Q, R, and C are all commutative rings under standard addition and multiplication. Z n is a commutative ring under addition mod n and multiplication mod n. For any n > 1, nz is a commutative ring without unity (under standard addition and multiplication). Z[x], the set of all polynomial expressions in x with coefficients in Z under ordinary addition and multiplication, is a commutative ring with unity (the constant polynomial p(x) = 1). M 2(Z), the set of all 2 2 matrices with entries in Z under matrix addition and multiplication, is a noncommutative ring with unity (namely, the identity matrix). C(R;a), the set of all continuous real-valued functions f(x) that vanish at x = a, under pointwise addition and multiplication, i.e., (f + g)(x) = f (x)+ g(x), (fg)(x) = f (x)g(x), is a commutative ring without unity. The set of elements of a ring that have multiplicative inverses are called the units of the ring. (What are the units in each of the rings above?)
12. Rings 4 Theorem If a, b, c are elements of the ring R, a0 = 0a = 0 a( b) = ( a)b = (ab) ( a)( b) = ab a(b c) = ab ac and (a b)c = ac bc If R has unity element 1, then ( 1)a = a. Theorem The unity element of a ring is unique, and if an element has a multiplicative inverse, this inverse is unique. As in the situation of a subgroup of a group, we call any subset of a ring that is itself a ring under the same two operations a subring. Theorem A nonempty subset S of a ring R is a subring of R if it is closed under subtraction and multiplication.
12. Rings 5 Examples: {0} is a subring of any ring; it is called the trivial subring nz is a subring of Z Z[i] = {a +bi a,b Z} is a subring of C called the ring of Gaussian integers the subset {0, 3, 6, 9, 12} is a subring of Z 15 ; while the unity in Z 15 is 1, the unity in this subring is 6 C(R;a) is a subring of the ring C(R) of all continuous real-valued functions f(x) under pointwise addition and multiplication Given rings R 1, R 2,, R n, we can combine them to form a larger ring, R 1 R 2 L R n = {(x 1, x 2,,x n ) x i R i }, where the addition and multiplication are both componentwise; this ring is called the direct sum of R 1, R 2,, R n.