As Drexel 28 said, the "unity" in a ring is the multiplicative identity. A "unit", on the other hand, is a member of the ring that has a multiplicative inverse (which, of course, is only possible in a ring with unity).

In

, the "unity" is 1 and numbers, n, for which (n, 10) (the least common multiple) is not 1 do not have inverses. Since 10= 2(5), the only possible "least common multiples" are 1, 2, 5, and 10. If (n, 10)= 2, then n= 2a so 5(n)= 5(2a)= 10a= 0 (mod 10). Similarly, if (n, 10)= 5, then n= 5a so 2(n)= 2(5a)= 10a= 0 (mod 10). Of course if (n, 10)= 10, n is itself equivalent to 0 (mod 10).

If mn= 0 with m and n both non-zero, in any ring, then n and m cannot have inverses- if n had an inverse,

then we would have

, a contradiction.