define what it means to say that R is a ring with a unit element?

Answer 1

A ring is a set R equipped with two binary operations
+ : R × R → R and
· : R × R → R, called addition and multiplication, such that:

(R, +) is an abelian group with identity element 0:
(1) (a + b) + c = a + (b + c)
(2) 0 + a = a + 0 = a
(3) a + b = b + a
(4) For every a in R, there exists an element denoted −a, such that a + −a = −a + a = 0

(R, ·) is a monoid with identity element 1:
(5) (a·b)·c = a·(b·c)
(6) 1·a = a·1 = a

Multiplication distributes over addition:
(7) a·(b + c) = (a·b) + (a·c)
(8) (a + b)·c = (a·c) + (b·c)

Rings need not have multiplicative inverses. An element a in a ring is called a unit if it is invertible with respect to multiplication: if there is an element b in the ring such that a·b = b·a = 1, then b is uniquely determined by a and we write a−1 = b. The set of all units in R forms a group under ring multiplication; this group is denoted by U(R) or R*.


*The above is quoted from the Wikipedia article. It's worth noting that some definitions of Ring don't require that property (6) hold, calling Rings that have a multiplicative identity unitary groups. Under this definition, it is possible that "unit element" in the question simply refers to the multiplicative identity, rather than some random unit, and hence the question is merely asking for the "standard" definition of a Ring.