My question is:

$R =\left\{\begin{pmatrix} a & b\\ c & d \end{pmatrix}\bigg|a,b,c,d \in\mathbb{R}\right\}$

Show that $T =\left\{\begin{pmatrix} a & 0\\ b & c \end{pmatrix}\bigg|a,b,c,d \in\mathbb{R}\right\}$ is a subring of R

Im not really sure how to prove this with my knowledge of rings.

Any help will be appreciated

  • 1
    $\begingroup$ What is the definition of a subring (hint: it's a subset of a ring that satisfies some properties)? You can show that $T$ is a subring of $R$ by showing that $T$ is a subset of $R$ that satisfies those properties. $\endgroup$ – ChocolateAndCheese Nov 10 '16 at 1:53
  • $\begingroup$ I already know the definition of a sub-ring i just dont know where to start in proving T is in R. @ChocolateAndCheese $\endgroup$ – user384716 Nov 10 '16 at 1:55
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    $\begingroup$ $R$ is the set of $2\times 2$ matrices with real entries. Is each element of $T$ a $2\times 2$ matrix with real entries? If so, then $T$ is a subset of $R$. $\endgroup$ – ChocolateAndCheese Nov 10 '16 at 1:57
  • $\begingroup$ Oh I see. So as R is the set of 2x2 matricies with real entities, any ring with a 2x2 matrix is a subset of R?? @ChocolateAndCheese $\endgroup$ – user384716 Nov 10 '16 at 2:03
  • $\begingroup$ The real question is "Is every element of $T$ also an element of $R$?" If so, then $T$ is a subset of $R$. TBH it sounds like it might be helpful for you to brush up on your set theory. Abstract algebra is a challenging subject, so it it critical to have a solid foundation. $\endgroup$ – ChocolateAndCheese Nov 10 '16 at 18:06

$1.$ Is $T$ non-empty?

$2.$ Is sum of two elements of $T$ in $T$?

$3.$ Is product of two elements of $T$ in $T$?

  • $\begingroup$ Isnt this just proving T is a ring?? $\endgroup$ – user384716 Nov 10 '16 at 1:56
  • $\begingroup$ HiHi..$T$ is a subring means $T$ is a ring only with the fact it is a subset of $R$ which it is already $\endgroup$ – Learnmore Nov 10 '16 at 1:59
  • $\begingroup$ @OPFragster Yes: A subring is basically a ring that is contained in another ring. (Just like a subset is a set contained in another set, or a subgroup is a group contained in another group). $\endgroup$ – ChocolateAndCheese Nov 10 '16 at 1:59
  • $\begingroup$ You also want $-x \in T$ for all $x \in T$ $\endgroup$ – SquirtleSquad Nov 10 '16 at 2:26

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