GRE Math Problem #57: Subring of R[x] From Real Numbers

[jammidactyl]

I'm reviewing the practice booklet for the GRE and came across a question I can't solve. Problem #57 for reference.

http://www.ets.org/Media/Tests/GRE/pdf/Math.pdf

Let R be the field of real numbers and R[x] the ring of polynomials in x with coefficients in R. Which of the following subsets of R[x] is a subring of R[x]?

I. All polynomials whose coefficient of x is zero.
II. All polynomials whose degree is an even integer, together with the zero polynomial.
III. All polynomials whose coefficients are rational numbers.

I figured the answer was "all of the above", but the answer in the back says just I and III.

If you add or subtract two polynomials of even degree, you get another polynomial of even degree or the zero polynomial. If you multiply two polynomials of even degree, the answer also is a polynomial of even degree. Since it's a subset and satisfies these conditions, isn't II a subring?

I think I'm making a really simple mistake with some obvious counterexample.

 

[d_leet]

Does II have a multiplicative identity?

 

[jammidactyl]

It only needs an additive identity to be a subring, which it has.

 

[jammidactyl]

Hrm... according to Wikipedia, some authors (of the books I happen to read) define separately "rings with unity". I guess I was just hung up on a matter of convention. Thanks!

http://en.wikipedia.org/wiki/Ring_(mathematics)

 

[jammidactyl]

Apparently it depends on which definition of the term "ring" you're used to! This is good information to know for the test... never realized there was such a difference.