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 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 ...
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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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### Proof Verification: Let S be a non-empty subset of a ring R. Then S is a subring of R if and only if S is closed under $-$ and $\times$.

Let S be a non-empty subset of a ring R. Then S is a subring of R if and only if S is closed under $-$ and $\times$. Proof: First, prove that S is a subgroup of R. Pick an arbitrary element $x$ from ...
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### Showing a subset of the functions $\mathbb R \to \mathbb R$ is a subring
Let $T$ be the set of all functions from $\mathbb R$ to $\mathbb R$. Let $S = \{ f \in T : f(2) = 0 \}$. Show that $S$ is a subring of $T$. $T$ has a zero element and an identity element by ...