I see that the definition of a subring requires that a subring should be closed under subtraction. But what I learned was that to be a subring, it should be closed under addition and multiplication. Where does that subtraction come from? Does it have something to do with inverse?
Asked
Active
Viewed 462 times
0
-
1Where did you get that from? – Chubby Chef Oct 23 '20 at 09:09
-
@ChubbyChef which one? the definition? – jun Oct 23 '20 at 09:09
-
Usually the procedure is to show separately that the subring is closed under addition and also that each element of the subring has an additive inverse. The answer below sums up the connection nicely. – Chubby Chef Oct 23 '20 at 09:14
-
@ChubbyChef I need to show additive inverse because of multiplication? Why do I need additive inverse? – jun Oct 23 '20 at 09:17
-
I guess this is from the definition of subring in chapter 7.1 Dummit and Foote. – Nabs Feb 14 '24 at 22:56
1 Answers
4
Being closed under addition and multiplication is not enough; after all, $\Bbb Z^+$ is closed under addition and multiplication, but it is not a subring of $\Bbb Z$. For each element $x$ of $S$, $-x$ must also belong to $S$. And if $S$ is closed under subtraction, then it is true that it is closed under addition and also that $x\in S\implies-x\in S$.
José Carlos Santos
- 427,504