Question

Question: Which of the following subsets of $R\left[X\right]$are sub rings of $R\left[X\right]$? Justify your answer:

1. All polynomials with constant term .
2. All polynomials of degree 2.
3. All polynomials of degrees $\le$$k$, where $k$is a fixed positive integer.
4. All polynomials in which the odd powers of $x$ have zero coefficients.
5. All polynomials in which the even powers of $x$ have zero coefficients.

Short answer

Answer:

The required answers are:

a. The following set is a subring:

b. It is not a subring as: $x^2·x^2=x^4$.

c. It is not a subring as: $x^k·x=x^{k+1}$.

d. It is a subring as:

e. It is not a subring as: $x·x=x^2$.

Step-by-step solution

Polynomial Arithmetic:

If any given function $R\left[x\right]$ is a ring, then the commutative, associative, and distributive laws hold such that the function $f\left(x\right)+g\left(x\right)$ exists.

Polynomials

(a)The subtraction and multiplication for all polynomials with constant terms , considered as non-empty subset can be expressed as:

Hence, the given set is a subring.

Polynomials:

(b)

The polynomials with degree 2 expressed as:

$x^2·x^2=x^4$

This violates the condition for a subring.

Hence, it is not a subring.

Polynomials:

(c)The polynomials of degree less than or equal to kexpressed as:

$x^k·x=x^{k+1}$

This violates the condition for a subring.

Hence, it is not a subring.

Polynomials:

(d) The subtraction and multiplication for all polynomials with odd powers of variables and zero coefficients considered as any non-empty subset can be expressed as:

Polynomials:

(e)The polynomials with even powers and zero coefficients can be expressed as:

$x·x=x^2$

This violates the condition for a subring.

Hence, it is not a subring.