text: 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
explanation: The document contains a math question and a partial discussion or attempt at answering the question. The user is asking about the properties of a specific set of matrices and whether it forms a subring and a field. The document also includes a response that clarifies the conditions for a subring and provides an example of matrix addition and multiplication.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: 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
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and a partial discussion or attempt at answering the question. The user is asking about the properties of a specific set of matrices and whether it forms a subring and a field. The document also includes a response that clarifies the conditions for a subring and provides an example of matrix addition and multiplication.
  
  Label: question-and-answer