mathbbZ2mathbbZ

mathbbZ2mathbbZ refers to the direct product of the ring of integers with itself, denoted as Z x Z. This mathematical structure is a set of ordered pairs of integers, where addition and multiplication are performed component-wise. For any two elements (a, b) and (c, d) in mathbbZ2mathbbZ, their sum is (a+c, b+d) and their product is (ac, bd).

This ring has several interesting properties. It is commutative, meaning that the order of multiplication does not matter: (a, b)(c, d) = (c, d)(a, b). It also has a multiplicative identity element, which is (1, 1). However, mathbbZ2mathbbZ is not an integral domain because it contains zero divisors. For example, (1, 0) is not the zero element, and (0, 1) is not the zero element, but their product (1, 0)(0, 1) is (0, 0), the zero element.

In terms of ideals, mathbbZ2mathbbZ has principal ideals. Any ideal in mathbbZ2mathbbZ can be generated by a single element. For instance, the ideal generated by (a, b) consists of all elements of the form k(a, b) where k is an integer. Understanding the structure of mathbbZ2mathbbZ is important in various areas of abstract algebra, including ring theory and module theory. It serves as a fundamental example of a direct product of rings.