Ideal (ring Theory) - Examples

Examples

• In a ring R, the set R itself forms an ideal of R. Also, the subset containing only the the additive identity 0_R forms an ideal. These two ideals are usually referred to as the trivial ideals of R.
• The even integers form an ideal in the ring Z of all integers; it is usually denoted by 2Z. This is because the sum of any even integers is even, and the product of any integer with an even integer is also even. Similarly, the set of all integers divisible by a fixed integer n is an ideal denoted nZ.
• The set of all polynomials with real coefficients which are divisible by the polynomial x2 + 1 is an ideal in the ring of all polynomials.
• The set of all n-by-n matrices whose last row is zero forms a right ideal in the ring of all n-by-n matrices. It is not a left ideal. The set of all n-by-n matrices whose last column is zero forms a left ideal but not a right ideal.
• The ring C(R) of all continuous functions f from R to R under pointwise multiplication contains the ideal of all continuous functions f such that f(1) = 0. Another ideal in C(R) is given by those functions which vanish for large enough arguments, i.e. those continuous functions f for which there exists a number L > 0 such that f(x) = 0 whenever |x| > L.
• Compact operators form an ideal in the ring of bounded operators.