<P> The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid . However, not every integer has a multiplicative inverse; e.g., there is no integer x such that 2x = 1, because the left hand side is even, while the right hand side is odd . This means that Z under multiplication is not a group . </P> <P> All the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity . It is the prototype of all objects of such algebraic structure . Only those equalities of expressions are true in Z for all values of variables, which are true in any unital commutative ring . Note that certain non-zero integers map to zero in certain rings . </P> <P> The lack of zero divisors in the integers (last property in the table) means that the commutative ring Z is an integral domain . </P> <P> The lack of multiplicative inverses, which is equivalent to the fact that Z is not closed under division, means that Z is not a field . The smallest field containing the integers as a subring is the field of rational numbers . The process of constructing the rationals from the integers can be mimicked to form the field of fractions of any integral domain . And back, starting from an algebraic number field (an extension of rational numbers), its ring of integers can be extracted, which includes Z as its subring . </P>

Lists and list t each contain 5 positive integers