Fundamental Constructs of Ring Theory

• A ring is a set with two binary operations and called addition and multiplication defined on such that the following hold

  • is an Abelian Group. ¹

  • Multiplication is associative

  • For all the left distributive law holds

    The right distributive law

    also holds

  • is the additive identity (which always exists because is a group)

    We denote the additive inverse of as

• (Fraleigh 18.8) If is a ring, then for any

• If for a ring a positive integer exists such that , then the least such positive integer is the characteristic of the ring. If no such positive integer exists, then has characteristic . We denote this

Misc §

• A subring of a ring is a subset of the ring that is a ring under induced operations from the whole ring. It is a generalization of Subgroup.

  • (Fraleigh e18.49a) The intersection of subrings of a ring is again a subring of

• An element of a ring is idempotent if .

• A ring is a Boolean Ring if every element is idempotent.

Links §

• Fraleigh

Footnotes §

1. The requirement that be Abelian is forced by distributivity. In particular, if we compute , commutativity is forced for both left distributivity and right distributivity to hold. ↩