Which is a subring?

  • #1

Main Question or Discussion Point

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 [Broken]

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.
 
Last edited by a moderator:

Answers and Replies

  • #2
1,074
1
Does II have a multiplicative identity?
 
  • #3
It only needs an additive identity to be a subring, which it has.
 
  • #4
  • #5
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.
 

Related Threads on Which is a subring?

  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
14
Views
4K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
4
Views
3K
  • Last Post
Replies
5
Views
3K
Replies
4
Views
4K
Replies
6
Views
2K
Replies
6
Views
2K
Replies
4
Views
3K
Replies
6
Views
951
Top