1. ## Rings

Consider the set of matrices S={[0 r, 0 0] | r is a real number} with normally defined addition and multiplication of matrices.

a) Show that S is a subring of M(R).
b) Show that S is not a field.

I know that for S to be a subring of M(R)is has to be closed under addition, but not multiplication. What does the normally defined addition and multiplication statement mean?

2. Originally Posted by empressA
Consider the set of matrices S={[0 r, 0 0] | r is a real number} with normally defined addition and multiplication of matrices.

a) Show that S is a subring of M(R).
b) Show that S is not a field.

I know that for S to be a subring of M(R)is has to be closed under addition, but not multiplication.
No, a subring must also be closed under multiplication. Saying that it is a ring rather than a field means that there may be elements other than 0 which do not have multiplicative inverses.

What does the normally defined addition and multiplication statement mean?
The nddition addition and multiplication of matrices, of course.

$\begin{bmatrix}0 & r \\ 0 & 0\end{bmatrix}+ \begin{bmatrix}0 & s \\ 0 & 0\end{bmatrix}= \begin{bmatrix}0 & r+s \\ 0 & 0\end{bmatrix}$

$\begin{bmatrix}0 & r \\ 0 & 0\end{bmatrix}\begin{bmatrix}0 & s \\ 0 & 0\end{bmatrix}$ = ?

3. ok...but when i multiply those matrices i end up with [0 0, 0 0]...would that still make it closed under multiplication..i know that it satisifes all the other requirements for a ring but am not sure if its closed under addition. Thats what I meant in the beginning I worded it wrong
The content structure consists of a discussion about a set of matrices and whether it forms a subring and a field, with some questions and clarifications provided.
no_question

the passage does not contain a direct mathematical question that requires a numerical or specific answer. instead, it is a discussion and explanation of concepts related to subrings and fields.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
1. ## Rings

Consider the set of matrices S={[0 r, 0 0] | r is a real number} with normally defined addition and multiplication of matrices.

a) Show that S is a subring of M(R).
b) Show that S is not a field.

I know that for S to be a subring of M(R)is has to be closed under addition, but not multiplication. What does the normally defined addition and multiplication statement mean?

2. Originally Posted by empressA
Consider the set of matrices S={[0 r, 0 0] | r is a real number} with normally defined addition and multiplication of matrices.

a) Show that S is a subring of M(R).
b) Show that S is not a field.

I know that for S to be a subring of M(R)is has to be closed under addition, but not multiplication.
No, a subring must also be closed under multiplication. Saying that it is a ring rather than a field means that there may be elements other than 0 which do not have multiplicative inverses.

What does the normally defined addition and multiplication statement mean?
The nddition addition and multiplication of matrices, of course.

$\begin{bmatrix}0 & r \\ 0 & 0\end{bmatrix}+ \begin{bmatrix}0 & s \\ 0 & 0\end{bmatrix}= \begin{bmatrix}0 & r+s \\ 0 & 0\end{bmatrix}$

$\begin{bmatrix}0 & r \\ 0 & 0\end{bmatrix}\begin{bmatrix}0 & s \\ 0 & 0\end{bmatrix}$ = ?

3. ok...but when i multiply those matrices i end up with [0 0, 0 0]...would that still make it closed under multiplication..i know that it satisifes all the other requirements for a ring but am not sure if its closed under addition. Thats what I meant in the beginning I worded it wrong
user
The content structure consists of a discussion about a set of matrices and whether it forms a subring and a field, with some questions and clarifications provided.

TYPE: NO_QUESTION

The passage does not contain a direct mathematical question that requires a numerical or specific answer. Instead, it is a discussion and explanation of concepts related to subrings and fields.