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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) Z_{n} = { 0, 1, . . . , n − 1}

with addition and multiplication mod n .

(Alternatively Z_{n} may be deﬁned to be the quotient ring Z/nZ , deﬁned below)

(5) R any ring, x an indeterminate. Put

R[[x]] = {a_{0} + a_{1}x + a_{2}x^{2} + . . . | a_{0}, a_{1}, . . . ∈ R} , the set of formal power series over R , which becomes a ring under addition and multiplication of power series. Important subring:

R[x] = {a_{0} + a_{1}x + . . . +a_{n}x^{n} | n ≥ 0 , a_{0}, a_{1}, . . . , a_{n} ∈ 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:

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 Z_{n} = {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 → Z_{n} where

f (z) = remainder after dividing z by n is a ring homomorphism with image Z_{n} and kernel nZ.

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