Is R a Subring of M2(Z)?

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Discussion Overview

The discussion centers on whether the set R, defined as matrices of the form [a a; b b] where a, b ∈ Z, is a subring of M2(Z). Participants explore the requirements for R to be considered a subring, including the existence of an identity element and closure under addition and multiplication.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant proposes that R needs to be shown to have an identity element to be a subring of M2(Z).
  • Another participant questions whether the identity element, if it exists, must be the same as that of M2(Z), noting that definitions of identity can vary among different ring theories.
  • There is a repeated inquiry about the conditions under which R can be considered a ring, specifically regarding closure under addition and multiplication.
  • Some participants suggest that the closure properties of R under addition and multiplication are critical to establishing it as a subring.
  • One participant mentions the subring test, implying that it could provide a framework for proving R's status as a subring.

Areas of Agreement / Disagreement

Participants express uncertainty regarding the identity element and its implications for R being a subring. There is no consensus on whether R meets the necessary criteria to be classified as a subring of M2(Z).

Contextual Notes

Participants have not fully resolved the assumptions regarding the identity element and the closure properties required for R to be a subring. The discussion reflects varying interpretations of what constitutes a ring, particularly concerning the identity element.

runescape09
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Let R = { martrix [a a (in the first row) b b (in the second row) | a,b∈Z }. Prove or disprove that R is a subring of M2(Z).

I've already know how to prove that R is the subring. But how do i show that their is an identity?
 
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if there IS an identity, wouldn't it have to be the same identity as M2(Z) has?

(note: some people require a ring to have identity, some people don't. by "identity" i mean multiplicative identity, as every ring MUST have a 0).
 
Deveno said:
if there IS an identity, wouldn't it have to be the same identity as M2(Z) has?

(note: some people require a ring to have identity, some people don't. by "identity" i mean multiplicative identity, as every ring MUST have a 0).

Okay, how do i know that the r is a ring?
 
is (R,+) closed under matrix addition? if A is in R, is -A in R? is R closed under matrix multiplication? these are the crucial questions.
 
Deveno said:
is (R,+) closed under matrix addition? if A is in R, is -A in R? is R closed under matrix multiplication? these are the crucial questions.

isn't that proven by the subring test though?
 

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