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

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In summary, the conversation discusses the definition of a subring in the context of R[x], the ring of polynomials with coefficients in the field of real numbers. The question asks which of the given subsets of R[x] is a subring, and the answer is given as only I and III, but the person speaking argues that II also satisfies the conditions to be a subring. The conversation also touches on the different definitions and conventions of a ring with unity.
  • #1
jammidactyl
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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.
 
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  • #2
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.
 

1. What is a subring?

A subring is a subset of a ring that is itself a ring. It contains a subset of the original ring's elements and follows the same operations and properties as the original ring.

2. How is a subring different from a ring?

A subring is a smaller version of a ring, while a ring is a more general mathematical structure. A ring can have multiple subrings, each with its own unique set of elements and properties.

3. What is R[x] in the context of this problem?

R[x] is the ring of polynomials with coefficients from the set of real numbers, R. It includes all polynomial expressions with real number coefficients, such as x^2 + 3x + 1.

4. How do you determine if a subset of R[x] is a subring?

To determine if a subset of R[x] is a subring, you must check if it follows the closure, associativity, identity, and inverse properties of a ring. In addition, it must also contain the additive and multiplicative identities of R[x] and be closed under addition and multiplication.

5. Can a subring of R[x] contain elements that are not polynomials?

No, a subring of R[x] must contain only polynomials with real number coefficients. Any elements that are not polynomials would not follow the same operations and properties as the rest of the subring's elements.

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