Rings, An Introduction

Introduction

If you're not familiar with groups, you really need to start here.  Rings are an extension of groups, and you should be familiar with normal subgroups and group homomorphisms in order to proceed with rings.

in fact a ring is a group and a monoid acting on the same set cooperatively.  The group implements something we call addition, and the monoid implements multiplication.  The integers form a ring; that's the classic example.  And now for the technical definition.

A ring is a set of elements S and two operators + and *.  The + operator implements an abelian group over S.  The * operator is associative, and distributes over +.  If * is commutative, we have a commutative ring.  The additive identity is denoted 0, and the multiplicative identity, if any, is denoted 1.  Unless stated otherwise, we will assume the ring contains 1, hence multiplication is at least a monoid.  If multiplication defines a group on the nonzero elements, i.e. everything other than 0 has a multiplicative inverse, the ring is a division ring.  The integers form a ring and the rationals form a division ring.  In fact the rationals form a commutative division ring, since multiplication commutes.  Such a ring is called a field.

For every x, 0*x = (1-1)*x = x-x = 0.  If 1 = 0 then x*1 = x*0 for every x, and the ring contains only 0, which isn't very interesting.  Henceforth we will assume 1 is different from 0.  The ring has at least two elements.  And there is a ring with precisely two elements, namely the integers mod 2, written Z2.

Continue adding 1+1+1+… until you reach 0.  If n becomes 0, then the ring has characteristic n.  If the multiples of 1 go on forever, the ring has characteristic 0.  Z (the integers) has characteristic 0, and Zn (the integers mod n) has characteristic n.

Semiring

A semiring is based on an additive monoid, rather than an additive group.  This is a bit of a misnomer.  You'd think a semiring would be based on a semigroup, but we really want the ring to contain 0, so addition is an abelian monoid.  The classic example is the nonnegative integers, or reals, under addition and multiplication.  The latter is a division semiring.

There is very little information on my website regarding semirings.  Many of the theorems on rings carry over to semirings; some do not.  You'll just have to step through them and see if the additive inverse is required.

If a semiring has characteristic n, it is automatically a ring.  Use the distributive property to show x + (n-1)x = 0.