# Homomorphic image of principal ideal ring

Question: show that homomorphic image of a principal ideal ring is a principal ideal ring.

My attempt : let $$R$$ be PIR (principal ideal ring) and $$f$$ be homomorphism from $$R$$ to a ring $$S$$ then by properties of ring homomorphism we know that $$f(R)$$ is subring of $$S$$. Now we consider an ideal $$I$$ of $$f(R)$$ then by properties of homomorphism we know its pullback that is $$f^{-1}(I)$$ is an ideal of $$R$$. But $$R$$ is PIR and hence there must exists some $$a\in R$$ such that $$f^{-1}(I)=$$.

Hence to prove the result we just need to show that $$I$$ is principal ideal. My intension is that $$I=$$. But how to prove it?

Let $$s\in $$ then by definition of principal ideal, $$s=s'f(a)$$ for some $$s'\in f(R)$$. How to show $$s\in I$$? If $$f(a)\in I$$ then as $$I$$ is an ideal of $$f(R)$$ hence we have $$s=s' f(a)\in I$$ and we are done! But how $$f(a)\in I$$? I didnt get this! (Since $$R$$ my not have unity and hence $$a$$ does not belongs to $$=f^{-1}(I)$$ and so we can't say $$f(a)\in I$$) and also how to show other direction that is, $$I\subseteq $$

• Is $<a>$ the smallest ideal in $R$ containing $a$? Or just $aR$?
– Plop
Apr 22, 2020 at 14:29
• @Plop sir just $aR$ Apr 22, 2020 at 14:30
• $aR = \{b \in R\ \vert \ \exists s, \ b = as\}$, right? So $x \in f(aR) \Leftrightarrow \exists c \in aR, \ x = f(c) \Leftrightarrow \exists b \in R, \ x = f(ab)=f(a)f(b) \Leftrightarrow x \in f(a)f(R)$.
– Plop
Apr 22, 2020 at 14:53
• Also, you seem to be using a nonstandard defintion of $\langle a\rangle$. The standard defintion guarantees $a\in\langle a\rangle$. Apr 23, 2020 at 15:32
• No, that is also an incorrect definition for the case of noncommutative rngs too. Are you saying you do want to consider noncommutativity? Why would you call $aR$ an ideal then? Are you saying that all right and left ideals are in fact principal? Apr 23, 2020 at 15:47

There seems to be a bit of ambiguity about what definitions you are using.

Since you make no mention of sides or commutativity, and you said $$aR$$ is an ideal, it looks like you are assuming commutativity.

$$\langle a\rangle$$ is, by definition (the standard definition), the smallest ideal of $$R$$ containing $$a$$, which would be $$aR+a\mathbb Z$$, in a commutative ring without identity.

Then from where you left off, I would continue that $$f(aR+a\mathbb Z)=f(a)f(R)+f(a)\mathbb Z=\langle f(a)\rangle$$ .

As far as I know, it is unheard of to call an ideal of the form $$xR$$ a principal ideal in a ring without identity. The problem is that you can never put your finger on a single element that generates $$xR$$ that way. (Pick any element $$xr$$ is $$xrR=xR$$? Who can tell?)

That's why we define $$\langle a\rangle$$ the other way so we can concretely name at least one element that generates it.

If you do in fact want to allow noncommutativity, you can use my same definition given above as the definition of a principle right ideal in a ring without identity. You'd have to make a corresponding one for principal left ideals. The arguments would still hold.

Finally, in case the solution to the actual problem wasn't clear yet, the strategy you are using should be the right one. The single generator of the pre-image of the ideal will map to a single generator of the ideal, proving the image is a principal ideal ring. All of these arguments continue to work if you are working with one-sided principal ideal rings.

• Sir, Thanks for your explanation and beautiful answer Apr 23, 2020 at 16:32
• Sir In your answer you written that "As far as I know, it is unheard of to call an ideal of the form $xR$ a principal ideal in a ring without identity." So is Wikipedia definition need some correction? Or it correct? Wikipedia define a left principal ideal of ring $R$ as $Rx$ and right principal ideal of ring $R$ as $xR$. (Even they didn't mention anything about $R$. That is they didn't mention that whether $R$ has unity or not ) Aug 30, 2021 at 15:21
• @AkashPatalwanshi No, wikipedia uses the convention of assuming identity. I'm sure there somewhere it has the "real" definition of principal ideals for rings without identity, but that will only be used in a narrow context. Aug 30, 2021 at 15:22
• sir, I read the wiki article carefully but i didn't find definition of principal ideal for rings without unity. Aug 30, 2021 at 15:25
• @AkashPatalwanshi Ok so the first thing I checked is the Wiki page for rings without identity and there it is. It’s the second bullet point. Aug 30, 2021 at 15:42