Linked Questions

28
votes
4answers
5k views

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 $a(a + 1) \mid m$.
5
votes
4answers
4k views

$a\mid b,\ c\mid d\,\Rightarrow\ ac\mid bd $ $\ \, \bf [Divisibility\ Product\ Rule]$

I just need to check the reasoning in my proof is correct, I think it is valid although I'm not totally convinced because I can't follow the logic; does proving that $x$ is an integer prove that $ac|...
4
votes
5answers
12k views

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 ...
2
votes
4answers
1k views

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 $F$ ...
1
vote
1answer
4k views

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 ...
12
votes
1answer
751 views

Prove that the ring $(\{0\},+,\cdot)$ is a subring of any ring $(R,+,\cdot).$

I have to prove that the ring $(\{0\},+,\cdot)$ is a subring of any ring $(R,+,\cdot).$ Let $S = (\{0\},+,\cdot)$ and $R = (R,+,\cdot)$ then $S$ is a subring of $R$ iff $(R,+,\cdot)$ is a ring and $S ...
0
votes
3answers
2k views

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 ...
2
votes
4answers
315 views

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 ...
6
votes
1answer
1k views

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 ...
2
votes
2answers
363 views

Proving that a subset of a ring $R$ is a subring

In this example, $R$ is a ring with unity $1$, with $a\in R$ having the property $a^2=a$ (making it a Boolean ring). I know every Boolean ring is of characteristic 2 since: $a+a=(a+a)^2=a^2+a^2+a^2+a^...
1
vote
1answer
370 views

Is $S = \{(a,b) \mid a + b = 0\}$ a subring of $\mathbb Z \times \mathbb Z$?

The homework question is "Is $S = \{(a,b) \mid a + b = 0\}$ a subring of $\mathbb Z \times \mathbb Z$? Justify your answer" Do I do this by checking the 8 axioms of a ring? If so how is the $\mathbb ...
1
vote
2answers
103 views

Is $R$ an integral domain?

Let $R = \{a + b\alpha |\ a,b \in \mathbb{Z}\}\subseteq \mathbb{C}$ where $\alpha = \frac{1}{2}(1+\sqrt{-19})$ Is $R$ an integral domain? To show whether or not $R$ is an integral domain, letting $x ...
-2
votes
1answer
291 views

Subring Test for non-empty subset $S ⊂ R$ [closed]

Show that a non-empty subset $S ⊂ R$ is a subring of $R$ if for all $r, s ∈ S$ we have $r − s ∈ S$ and $rs ∈ S$. (This makes it easier to verify a set is a ring, if you know the set lives in a larger ...
1
vote
3answers
147 views

Prove that $\{x + y \cdot \sqrt{3} | x, y \in \mathbb{Z} \}$ is a ring (or not)

How to prove that $(R, +, \cdot)$ is a ring (or not), where $R = \{x + y \cdot \sqrt{3} | x, y \in \mathbb{Z} \}$? Update. Is this proof correct? $(R, +)$ is an abelian group: Closure: $a, b \in R \...
1
vote
2answers
98 views

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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