Rings
A ring A is a set, together with two laws of composition called multiplication and addition respectively, and written as a product and as a sum respectively, satisfying the following conditions: RI 1. With respect to addition, A is a commutative group. R
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II
Rings
§1.
RINGS AND HOMOMORPHISMS
A ring A is a set, together with two laws of composition called multiplication and addition respectively, and written as a product and as a sum respectively, satisfying the following conditions:
RI 1. With respect to addition, A is a commutative group. RI 2. The multiplication is associative, and has a unit element.
RI3. For all x, y, (x
ZE
+ y)z =
A we have xz
+ yz
and
z(x
+ y) = zx + zy.
(This is called distributivity.) As usual, we denote the unit element for addition by 0, and the unit element for multiplication by 1. We do not assume that I "# O. We observe that Ox = 0 for all x E A. Proof : We have Ox + x = (0 + I)x = lx = x. Hence Ox = O. In particular, if I = 0, then A consists of 0 alone. For any x, yEA we have (-x)y = -(xy). Proof: We have xy
+ (-x)y = (x + (-x))y = Oy = 0,
so (-x)y is the additive inverse of xy. Other standard laws relating addition and multiplication are easily proved, for instance (- x)( - y) = xy . We leave these as exercises. Let A be a ring, and let V be the set of elements of A which have both a right and left inverse. Then V is a multiplicative group. Indeed, if a has a
83 S. Lang, Algebra © Springer Science+Business Media LLC 2002
84
II. §1
RINGS
right inverse b, so that ab = 1, and a left inverse c, so that ca = 1, then cab = b, whence c = b, and we see that c (or b) is a two-sided inverse, and that c itself has a two-sided inverse, namely a. Therefore U satisfies all the axioms of a multiplicative group, and is called the group of units of A. It is sometimes denoted by A*, and is also called the group of invertible elements of A. A ring A such that 1 # 0, and such that every non-zero element is invertible is called a division ring. Note. The elements of a ring which are left invertible do not necessarily form a group. Example. (The Shift Operator).
Let E be the set of all sequences
a = (a1 , a2 , a3" ") of integers. One can define addition componentwise. Let R be the set of all mappings f : E -. E of E into itself such that f(a + b) = f(a) + f(b). The law of composition is defined to be composition of mappings. Then R is a ring. (Proof?) Let
Verify that T is left invertible but not right invertible. A ring A is said to be commutative if xy = yx for all x, yEA. A commutative division ring is called a field. We observe that by definition, a field contains at least two elements, namely 0 and 1. A subset B of a ring A is called a subring if it is an additive subgroup, if it contains the multiplicative unit, and if x, y E B implies xy E B. If that is the case, then B itself is a ring, the laws of operation in B being the same as the laws of operation in A. For example, the center of a ring A is the subset of A consisting of all elements a E A such that ax = xa for all x E A. One sees immediately that the center of A is a subring. Just as we proved general associativity from the associativity for three factors, one can prove general distributivity. If x , Yl ' ... , Yn are elements of a ring A, then by induction one sees t
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