17 questions linked to/from The subring test
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### If $a \mid m$ and $(a + 1) \mid m$, prove $a(a + 1) | m$.

Can anyone help me out here? Can't seem to find the right rules of divisibility to show this: If $a \mid m$ and $(a + 1) \mid m$, then $a(a + 1) \mid m$.
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### Prove the following is a subring

Let $R=\{m+n\sqrt{2} \mid m,n\in \mathbb{Z}\}$. Prove that $R$ is a subring of the real numbers. I just want to know how to get started really. My professor has used the same example for the past ...
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### Show that $A=\{x+y\sqrt{2}:x,y \in \mathbb{Z}\}$ is a commutative ring with unity, find the zero element, the unity and the negative of $a$

Show that $A=\{x+y\sqrt{2}:x,y \in \mathbb{Z}\}$ is a commutative ring with unity, find the zero element, the unity and the negative of an arbitrary $a$. First thing first, I need to show it is a ...
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### Subring of F[x], all with linear term = 0, is not a UFD

My question pertains to this link (the content of which has been included below in the most recent edit) The ring of polynomials over a field with no linear term is not a UFD Let $F$ ...
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### How to prove a set is an integral domain?

Let's say I have the ring: $\mathbb{Z}[\sqrt{2}]=\{a+b\sqrt{2}\mid a,b\in\mathbb{Z}\}$. Now the question asked is to prove whether or not this ring is an integral domain. By definition: "An ...
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### Prove that the center of a ring is a subring.

The center of a ring $R$ is $\{c\in R : cr=rc$ for every $r \in R\}$. Prove that the center of a ring is a subring. What is the center of a commutative ring? Is my solution right? solution You ...
### Prove that $(n\mathbb Z, +, \times )$ are the only subrings of $(\mathbb Z, +, \times)$
I had to find all the subrings of the integers and then prove that there aren't any more. It's clear to me the $(n\mathbb Z, +, \times )$ is a subring of the integers for all $n$ element of the ...