ideal (idea)

A right ideal I of a ring R is a nonempty subset of R such that

• a+b in I for a,b in I
• I is closed under right multiplication. i.e. for each a in I and r in R we have ar in I.

A left ideal is defined similarly, except that it has to be closed under left multiplication.

A two-sided ideal (or just ideal) is a right and left ideal of a ring.

For commutative rings there is no difference between the concepts of right,left and two-sided ideals.

Examples of ideals

• If R is any ring then R is itself an ideal of R. So is {0}.
• If a is an element of R then

  aR = {ar: r in R}

  is a right ideal of R. More generally if a₁,...,a_n are finitely many elements of R then

  a₁R +...+ a_nR = {a₁r₁+...+a_nr_n: each r_i in R}

  is a right ideal of R.
• Let Z be the ring of integers. Then nZ (i.e. all integer mutiples of n) is an ideal.
• Let x,y be complex numbers then the set of all 2x2 complex matrices of the form

   --   --
  | xa xb |
  | yc yd |
   --   --
  

  with a,b,c,d complex numbers, form a right ideal of the ring of complex 2x2 matrices.

If I and J are right ideals then so is I+J which consists of all sums i+j with i in I and j in J. If I and J are ideals then so is IJ which consists of all finite sums i₁j₁+...+i_kj_k, with i_r in I and j_s in J. If S is a subset of R the right ideal it generates is denoted by SR and consists of all finite sums i₁j₁+...+i_kj_k, with i_r in S and j_s in R. Note that SR is a right ideal of R. Finally, if I_j is a family of right ideals then the sum of this family consists of all finite sums i_j1 +...+ i_j1 with i_jk in I_jk. The sum is again a right ideal. If a is an element of R then the ideal it generates RaR consists of all finite sums r₁as₁ + ... + r_tas_t with r_i and s_i in R. This is an ideal. If S is a subset of R then the ideal of R it generates is the sum of the ideals RaR, with a in S. We sometimes write (a₁,...,a_n) for the ideal generated by {a₁,...,a_n}.