curiousmuch said:thanks, but we can't assume R is closed under multiplication.
A finite commutative ring is a set of elements with two operations, addition and multiplication, that follow certain rules. The ring is finite, meaning it has a limited number of elements, and it is commutative, meaning the order of operations does not affect the result.
An integral domain is a ring in which every element has a unique multiplicative inverse. This means that every non-zero element in the ring has a corresponding element that, when multiplied together, result in the multiplicative identity element (usually 1).
To prove that a finite commutative ring is an integral domain, one must show that every element in the ring has a unique multiplicative inverse, and that there are no zero divisors. This can be done by examining the properties of the ring's operations and showing that they adhere to the rules of an integral domain.
Zero divisors are elements in a finite commutative ring that, when multiplied together, result in the additive identity element (usually 0). In other words, they are elements that have a product of 0, but are not themselves equal to 0.
Having no zero divisors is important because it ensures the ring is an integral domain. This means that every element has a unique multiplicative inverse, allowing for more complex mathematical operations, and also prevents errors or inconsistencies in calculations.