Units of Ring Direct Product are Ring Direct Product of Units

Theorem

Let $R_1, R_2, \ldots, R_m$ be rings with identity.

This article needs to be linked to other articles.
In particular: group isomorphism? This is rings we're talking about here.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{MissingLinks}} from the code.

Then the following group isomorphism holds:

$\map U {R_1 \oplus R_2 \oplus \cdots \oplus R_n} \simeq \map U {R_1} \oplus \map U {R_2} \oplus \cdots \oplus \map U {R_n}$.

This article, or a section of it, needs explaining.
In particular: what's $U$? What's $\oplus$?
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Explain}} from the code.

Proof

It suffices to prove the case $m = 2$.

This article, or a section of it, needs explaining.
In particular: no it doesn't, we need an inductive argument
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Explain}} from the code.

By Definition of Ring Direct Sum the finite direct sum

This article, or a section of it, needs explaining.
In particular: Is ring direct sum the same thing as finite direct sum? At least be consistent.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Explain}} from the code.

is isomorphic to the finite direct product.

There is believed to be a mistake here, possibly a typo.
In particular: I believe the wrong link has been implemented, because it does not match the text. Is this the RDP being linked to? Linking is of paramount importance on this site.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by reviewing it, and either correcting it or adding some explanatory material as to why you believe it is actually correct after all.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Mistake}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

$\ds \tuple {r_1, r_2}$

$\in$

$\ds \map U {R_1 \times R_2}$

Definition of Unit of Ring

$\ds \leadstoandfrom \ \ $

$\ds \tuple {r_1, r_2}$

$\ds $

is invertible in $R_1 \times R_2$

$\ds \leadstoandfrom \ \ $

$\ds \exists \tuple {s_1, s_2} \in R_1 \times R_2: \, $

$\ds \tuple {r_1, r_2} \tuple {s_1, s_2}$

$=$

$\ds \tuple {s_1, s_2} \tuple {r_1, r_2}$

$\ds = \tuple {1_{R_1}, 1_{R_2} }$

$\ds \leadstoandfrom \ \ $

$\ds \tuple {r_1 s_1, r_2 s_2}$

$=$

$\ds \tuple {s_1 r_1, s_2 r_2}$

$\ds = \tuple {1_{R_1}, 1_{R_2} }$

$\ds \leadstoandfrom \ \ $

$\ds r_1 s_1$

$=$

$\ds s_1 r_1$

$\ds = 1_{R_1}$

$\, \ds \land \, $

$\ds r_2 s_2$

$=$

$\ds s_2 r_2$

$\ds = 1_{R_2}$

$\ds \leadstoandfrom \ \ $

$\ds r_1$

$\ds $

is invertible in $R_1$

$\, \ds \land \, $

$\ds r_2$

$\ds $

is invertible in $R_2$

$\ds \leadstoandfrom \ \ $

$\ds r_1$

$\in$

$\ds \map U {R_1}$

Definition of Unit of Ring

$\, \ds \land \, $

$\ds r_2$

$\in$

$\ds \map U {R_2}$

Definition of Unit of Ring

$\blacksquare$

Units of Ring Direct Product are Ring Direct Product of Units

Theorem

Proof

Navigation menu

Search