integral domain
An integral domain is a ring that behaves like the integers in one decisive respect: you can never multiply two nonzero things and get zero. This is the property that lets you cancel — if a · b = a · c and a is not zero, you may safely conclude b = c, exactly as you expect with ordinary numbers. Many rings lack this comfort.
Formally, an integral domain is a commutative ring with a multiplicative identity 1 (different from 0) that has no zero divisors: if a · b = 0, then a = 0 or b = 0. A zero divisor is a nonzero element that multiplies with another nonzero element to give zero, and integral domains are precisely the commutative rings free of them.
Integral domains sit neatly between general commutative rings and fields. Every field is an integral domain, but not conversely — the integers are the prototypical integral domain yet famously not a field. Where they fail to be fields is exactly the absence of multiplicative inverses; otherwise they are as well-behaved as arithmetic gets.
ℤ is an integral domain: no two nonzero integers multiply to 0. But the integers mod 6 are not, because 2 · 3 = 6 ≡ 0 even though neither 2 nor 3 is zero — here 2 and 3 are zero divisors.
ℤ has no zero divisors; ℤ/6ℤ has 2·3 = 0.