A ring A is a set with + , • such that
(1) (A, +) is an abelian group; (2) (A, •) is a semigroup; (3) • distributes over + on both sides. (4) (∀x, y ∈ A) x • y = y • x (5) (∃1 ∈ A)(∀x ∈ A) 1•x = x•1 = x . |
In this course all rings A are commutative, that is, and have an identity element 1 (easily seen to be unique)
If 1 = 0 then A = {0} (easy to see), called the zero ring. |
Multiplication will be denoted by juxtaposition, and simple facts used without comment, such as
(∀x, y ∈ A) x 0 = 0 , (−x)y = x(−y) = −(xy) , (−x)(−y) = xy |
Call a subset S of a ring A a subring if
(i) 1 ∈ S ; (ii) (∀x, y ∈ S ) x + y , xy , −x ∈ S . |
Condition (ii) is easily seen to be equivalent to
(ii)′ (∀x, y ∈ S ) x − y , xy ∈ S . |
Note: In other contexts authors replace the condition 1 ∈ S by S ≠ ∅ (which is not equivalent!). |
Examples:
(1) Z is the only subring of Z .
(2) Z is a subring of Q , which is a subring of R , which is a subring of C .
(3) Z[i] = { a + bi | a, b ∈ Z }
the ring of Gaussian integers is a subring of C
(4) Zn = { 0, 1, . . . , n − 1}
with addition and multiplication mod n .
(Alternatively Zn may be defined to be the quotient ring Z/nZ , defined below)
(5) R any ring, x an indeterminate. Put
R[[x]] = {a0 + a1x + a2x2 + . . . | a0, a1, . . . ∈ R} , the set of formal power series over R , which becomes a ring under addition and multiplication of power series. Important subring:
R[x] = {a0 + a1x + . . . +anxn | n ≥ 0 , a0, a1, . . . , an ∈ R}
the ring of polynomials over R .
Call a mapping f : A → B (where A and B are rings) a ring homomorphism if
(a) f (1) = 1 ; (b) (∀x, y ∈ A) f (x + y) = f (x) + f (y) |
in which case the following are easily checked:
(i) f (0) = 0 ; |
An isomorphism is a bijective homomorphism, say f : A → B , in which case we write
A ≌ B or f : A ≌ B.
It is easy to check that
≌ is an equivalence relation. |
A nonempty subset I of a ring A is called an ideal, written
(i) (∀x, y ∈ I ) x + y , −x ∈ I [clearly equivalent to (i)′ (∀x, y ∈ I ) x − y ∈ I ]: (ii) (∀x ∈ I )(∀y ∈ A) xy ∈ I . |
In particular I is an additive subgroup of A , so we can form the quotient group the group of cosets of I ,
A/I = { I + a | a ∈ A} ,
with addition defined by, for a, b ∈ A , (I + a) + (I + b) = I + (a + b) .
Further A/I forms a ring by defining, for a, b ∈ A ,
(I + a) (I + b) = I + (a b) .
Verification of the ring axioms is straightforward: |
— only tricky bit is first checking multiplication is well-defined:
If I + a = I + a′ and I + b = I + b′ then
a − a′ , b − b′ ∈ I ,
so
ab − a′b′ = ab − ab′ + ab′ − a′b′
= a(b − b′) + (a − a′)b′ ∈ I
yielding I + ab = I + a′b′ .
We call A/I a quotient ring.
The mapping
φ : A → A/I , x → I + x
is clearly a surjective ring homomorphism, called the natural map, whose kernel is
ker φ = {x ∈ A | I + x = I} = I .
Thus all ideals are kernels of ring homomorphisms.
The converse is easy to check, so
kernels of ring homomorphisms with domain A are precisely ideals of A . |
The following important result is easy to verify:
Fundamental Homomorphism Theorem: If f : A → B is a ring homomorphism with kernel I and image C then A/I ≌ C . |
Proposition: and φ : A → A/I be the natural map. Then(i) ideals J of A/I have the form J = J/I = {I + j | j ∈ J} for some J such that I ⊆ J A ; (ii) φ−1 is an inclusion-preserving bijection between ideals of A/I and ideals of A containing I . |
Example: The ring Zn = {0, 1, . . . , n − 1} with mod n arithmetic is isomorphic to Z/nZ : follows from the Fundamental Homomorphism
Theorem, by observing that the mapping f : Z → Zn where
f (z) = remainder after dividing z by n is a ring homomorphism with image Zn and kernel nZ.
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1. What is a ring in mathematics? |
2. What are ideals in ring theory? |
3. What is a quotient ring in ring theory? |
4. How are ideals related to quotient rings? |
5. What are some applications of ring theory in mathematics? |
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