1.1 Rings

A ring is a set RR with two binary operations ++ and ⋅\cdot such that

  1. (a)
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    (R,+)(R,+) is a commutative group;

  2. (b)
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    ⋅\cdot is associative, and there exists11 1 We require that rings have a 11, which entails that we require homomorphisms to preserve it. an element 1R1_{R} such that a⋅1R=a=1R⋅aa\cdot 1_{R}=a=1_{R}\cdot a for all a∈R;a\in R;

  3. (c)
    ​

    the distributive law holds: for all a,b,c∈Ra,b,c\in R,

    (a+b)⋅c\displaystyle(a+b)\cdot c
    =a⋅c+b⋅c\displaystyle=a\cdot c+b\cdot c
    a⋅(b+c)\displaystyle a\cdot(b+c)
    =a⋅b+a⋅c⁢.\displaystyle=a\cdot b+a\cdot c\text{.}

We usually omit “⋅\cdot” and write 11 for 1R1_{R} when this causes no confusion. If 1R=01_{R}=0, then R={0}R=\{0\}.

A subring of a ring RR is a subset SS that contains 1R1_{R} and is closed under addition, passage to the negative, and multiplication. It inherits the structure of a ring from that on RR.

A homomorphism of rings α:R→R′\alpha\colon R\rightarrow R^{\prime} is a map such that

α⁢(a+b)=α⁢(a)+α⁢(b),α⁢(a⁢b)=α⁢(a)⁢α⁢(b),α⁢(1R)=1R′\alpha(a+b)=\alpha(a)+\alpha(b),\quad\alpha(ab)=\alpha(a)\alpha(b),\quad\alpha% (1_{R})=1_{R^{\prime}}

for all a,b∈Ra,b\in R. A ring RR is said to be commutative if multiplication is commutative:

a⁢b=b⁢a⁢ for all ⁢a,b∈R.ab=ba\text{ for all }a,b\in R.

A commutative ring is said to be an integral domain if 1R≠01_{R}\neq 0 and the cancellation law holds for multiplication,

a⁢b=a⁢c⁢, ⁢a≠0⁢, implies ⁢b=c.ab=ac\text{, }a\neq 0\text{, implies }b=c.

An ideal II in a commutative ring RR is a subgroup of (R,+)(R,+) that is closed under multiplication by elements of RR,

r∈R⁢, ⁢a∈I⁢, implies ⁢r⁢a∈I.r\in R\text{, }a\in I\text{, implies }ra\in I.

The ideal generated by elements a1,…,ana_{1},\ldots,a_{n} is denoted by (a1,…,an)(a_{1},\ldots,a_{n}). For example, (a)(a) is the principal ideal a⁢RaR.

We assume that the reader has some familiarity with the elementary theory of rings. For example, in ℤ\mathbb{Z}{} (more generally, any Euclidean domain) an ideal II is generated by any “smallest” nonzero element of II, and unique factorization into powers of prime elements holds. We write gcd⁡(a,b)\gcd(a,b) for the greatest common divisor of aa and bb, e.g., gcd⁡(a,0)=a.\gcd(a,0)=a.