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I know that $2\mathbb{N}$ or $2\mathbb{N}+1$ are notations for even and odd naturals.

  1. What kind of set (builder?) notation is $5\mathbb{Z}+3$? I know it is something like {...,-2,3,8,18,...} intuitively but i am on the notation. I don't think that we could use any notation arbitrarily that comes to our mind.
  2. Are there any standards or name (or book reference) for this kind of set definition?
  3. Is it derived from or related to ideal of ring $\mathbb{Z}$?
  4. Also, could write with this form arbitrarily like $6\mathbb{Z}+5\mathbb{Z}+7$?
Asaf Karagila
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lockedscope
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    yes it basically works as you said, but keep in mind that $a\mathbb Z+b\mathbb Z=\gcd(a,b)\mathbb Z$ for the additive group so the last one is just plain $\mathbb Z$. – zwim Oct 04 '20 at 22:30
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    This object is called a "coset" - a quick google will give you relevant resources. – Mummy the turkey Oct 04 '20 at 22:32
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    I think it goes good, but a better way would be to write "Let us define the set $5\mathbb{Z}+3$ containing all integers of the form $5x+3$ for all $x\in\mathbb{Z}$". So, in case the reader is not aware about the notation, he can still understand it afterwards. – ultralegend5385 Oct 05 '20 at 04:26

2 Answers2

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This notation is a combination of standard notation for a subgroup and a coset.

First $(5\mathbb{Z},+)$ is an additive subgroup$(\{...,-10,-5,0,5,10,...\},+)$. Then $5\mathbb{Z}+3$ is a coset: a set that we got by adding $3$ to each element of $5\mathbb{Z}$.

Technically it is not clear what would $a\mathbb{Z}+b\mathbb{Z}+c$ mean because first + sign and second + sign must have a completely different meaning. So I am objecting to such usage.

Aleksandar
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  • I always thought the notation $n\mathbb{Z}$ comes from the cosets of the semigroup $(\mathbb{Z}, \cdot)$ – Stefan Octavian Oct 05 '20 at 11:10
  • @StefanOctavian No, that does not make a lot of sense; $n\mathbb{Z}$ would be itself a coset of what substructure? And how then would $n\mathbb{Z}+k$ be interpreted in this? – xxxxxxxxx Oct 05 '20 at 16:39
  • Well, I thought $n\mathbb{Z}$ would be a coset in $(\mathbb{Z}, \cdot)$ of itself as a(/an)(improper) subsemigroup. And then, with the notation established for the set $n\mathbb{Z}$ and seeing that it's a subgroup of $(\mathbb{Z}, +)$, we build cosets $n\mathbb{Z} + k$. I know that doesn't make much sense, it was just a try at explaining the notation to myself. – Stefan Octavian Oct 05 '20 at 17:35
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Let $R$ be a ring, $A,B \subseteq R,$ and $c \in R$. Then we define $$ A+B = \{ a+b \mid a \in A,\ b \in B \} \\ cA = \{ ca \mid a \in A \} \\ A+c = \{ a+c \mid a \in A \} $$

Thus, for example, $$ 5 \mathbb{Z} + 3 = \{ 5n+3 \mid n \in \mathbb{Z} \}. $$

md2perpe
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