Ideal

• An additive Subgroup of a ring is called an Ideal if it satisfies

  This is analogous to the formulation for normal groups.

• Every nonzero ring has at least two ideals — the improper ideal and the trivial ideal

• (Fraleigh e26.22) Let be an ideal of and a Ring Homomorphism. Then is an ideal of although it need not be an ideal of

  Also if is an ideal of either or , then is also an ideal of .

• (Fraleigh e26.27) The intersection of ideals of ring is also an ideal of .

• A maximal ideal of a ring is an ideal such that there is no proper ideal of properly containing .

• (Fraleigh 27.9) Let be a Commutative Ring with unity. Then is a maximal ideal of if and only if is a field

• An ideal in commutative ring is a prime ideal if

  • The motivation behind this definition is that the factor ring will be an Integral Domain if and only if
    That is has no divisors of . See more in (Fraleigh 27.15)

• (Fraleigh 27.16) Every maximal ideal in a commutative ring with unity is a prime ideal.

• (Fraleigh e27.24) For a finite commutative ring with unity , every prime ideal is a maximal ideal.

• If is a commutative ring with unity and , the ideal of all multiples of is called the principal ideal generated by defined as

  An ideal is called a principal ideal if for some . That is it is an ideal generated by exactly one element

  • Intuition: We can think of a principal ideal as the set of “multiples” of
  • Every ideal of is a principal ideal.
  • If we have then we denote the ideal generated by these as

Topics §

• Ideal Arithmetic
• Groebner Basis
• Factor Ring

Links §

• Fraleigh