2. Define a set X as X = RU {-}. Define two binary operations over X as follows: Addition:x O y = max (x, y) Multiplication: xy = x + y (here "+" denotes the usual addition or real numbers) investigate the properties of a commutative ring for this structure; that is, write proofs for the properties that hold or give an explanation on why a property does not hold if it doesn't What kind of algebraic structure is this? Note: The properties of -oare the same as what you would have seen in Calculus courses.

13. Suppose that a and b belong to a commutative ring and ab is a zero divisor. Show that either a or b is a zero divisor.