Think of it in terms of numbers, of a certain type, like real or integer for example, with operations defined on them.
Say, the integers with addition. Subtraction isn't defined except where you add a negative integer. The negative integers are all defined as additive inverses of their corresponding positive integers.
But with multiplication defined on integers you immediately have a problem with inverses. Hence the integers are said to not be closed under multiplication; the multiplicative inverses of integers in fact divide other integers.
However, the ring of integers does have addition and multiplication defined. A ring is generally an Abelian group under addition (and additive inverses exist), which also has an associative multiplication that is left and right distributive over addition (and multiplicative inverses need not exist).
A ring with multiplicative inverses for all non-zero elements is called a division ring. Note that multiplication by zero has no inverse. (another way of saying multiplication by $$ 0^{-1} $$ is not defined).
Say, the integers with addition. Subtraction isn't defined except where you add a negative integer. The negative integers are all defined as additive inverses of their corresponding positive integers.
But with multiplication defined on integers you immediately have a problem with inverses. Hence the integers are said to not be closed under multiplication; the multiplicative inverses of integers in fact divide other integers.
However, the ring of integers does have addition and multiplication defined. A ring is generally an Abelian group under addition (and additive inverses exist), which also has an associative multiplication that is left and right distributive over addition (and multiplicative inverses need not exist).
A ring with multiplicative inverses for all non-zero elements is called a division ring. Note that multiplication by zero has no inverse. (another way of saying multiplication by $$ 0^{-1} $$ is not defined).