Download Rings Many of the groups with which we are familiar are arithmetical

12. Rings
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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
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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
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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.
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Examples:
• Naturally, Z, Q, R, and C are all commutative
rings under standard addition and
multiplication.
• Zn 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).
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• 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?)
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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.
Proof Easy. //
Theorem The unity element of a ring is unique,
and if an element has a multiplicative inverse, this
inverse is unique.
Proof Easy. //
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.
Proof Easy. //
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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 Z15 ; while
the unity in Z15 is 1, the unity in this subring is 6
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• C(R;a) is a subring of the ring C(R) of all
continuous real-valued functions
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pointwise addition and multiplication
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Given rings R1, R2 ,…, Rn , we can combine them to
form a larger ring,
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R1 ⊕ R2 ⊕ L ⊕ Rn = {(x1, x2,…, x n )|xi ∈ Ri },
where the addition and multiplication are both
componentwise; this ring is called the direct sum
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of R1, R2 ,…, Rn .
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