that is not the unit ideal is called a proper ideal (as it is a proper subset). Note: a left ideal is proper if and only if it does not contain a unit element, since if is a unit element, then for every . Typically there are plenty of proper ideals. In fact, if R is a skew-field, then are its only ideals and conversely: that is, a nonzero ring R is a skew-field if are the only left (or right) ideals. (Proof: if is a nonzero element, then the principal left ideal (see below) is nonzero and thus ; i.e., for some nonzero . Likewise, for some nonzero . Then .) The even integers form an ideal in the ring of all integers; it is usually denoted by . 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 . 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 of all continuous functions f from to under pointwise multiplication contains the ideal of all continuous functions f such that f(1) = 0. Another ideal in 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 > L. A ring is called a simple ring if it is nonzero and has no two-sided ideals other than . Thus, a skew-field is simple and a simple commutative ring is a field. The matrix ring over a skew-field is a simple ring. If is a ring homomorphism, then the kernel is a two-sided ideal of . By definition, , and thus if is not the zero ring (so ), then is a proper ideal. More generally, for each left ideal I of S, the pre-image is a left ideal. If I is a left