Understanding a corollary of the universal property of tensor products of modules

This is a follow-up question of a previous one of mine.

The following theorem is from the Abstract Algebra by Dummit and Foote (in the section 10.4 tensor products of modules): Here are my questions:

• What does "largest quotient of $N$" mean in the proof? (Being "largest" in what sense?)
• How does one get "It follows that $N/\hbox{ker}\iota$ is the unique largest quotient of $N$ that can be embedded in any $S$-module"?
• Would anyone elaborate how "the last statement in the corollary follows immediately"?

Here is the reference for Theorem 8 mentioned above. (1) I take the definition of "largest quotient" to be the sentence before. I.e., if $N/\ker(\varphi)$ is another quotient of $N$ that can be embedded $N/\ker(\varphi) \hookrightarrow L$ into an $S$-module $L$, then there is a unique $R$-linear surjection $\pi: N/\ker(\iota) \to N/\ker(\varphi)$ such that the following diagram commutes. (If $A$ surjects onto $B$, I usually think of $A$ as being bigger, which is at least true in terms of cardinality.)

As mentioned in the corollary, this realizes $N/\ker(\varphi)$ as a quotient of $N/\ker(\iota)$: $$N/\ker(\varphi) \cong \frac{N/\ker(\iota)}{\ker(\varphi)/\ker(\iota)} \, .$$

You can also think dually, in terms of ideals. $N/\ker(\iota)$ being the largest means that $\ker(\iota)$ is the smallest: if $\varphi: N \to L$ is an $S$-module homomorphism, then $\ker(\iota) \subseteq \ker(\varphi)$. This is because we "throw out" the least when quotienting by the smallest possible ideal.

Now that I think about it, I think the authors mean "largest" in terms of posets. The ideals that induce embeddings form a poset under inclusion. If we instead look at the quotients by these ideals, we get a "dual" poset where the ordering is reversed, as inclusion maps are replaced by quotient maps.

(2) now just follows from the definition of "largest."

(3) If $\iota$ is injective, then $\ker(\iota) = 0$, so $N/\ker(\iota) \cong N$ injects into $S \otimes_R N$, which is an $S$-module. Conversely, suppose that $\varphi: N \hookrightarrow L$ is an injection of $N$ into an $S$-module $L$. Then there exists a $\Phi$ as in the theorem such that $\varphi = \Phi \circ \iota$, so the injectivity of $\varphi$ implies that $\Phi$ and $\iota$ must both be injective, too.

Actually, I guess we can appeal directly to the corollary for the reverse implication. Suppose $\varphi: N \to L$ is an injection, so $\ker(\varphi) = 0$. Then we get a map $\pi: N/\ker(\iota) \to N/\ker(\varphi)$, so we must have $\ker(\iota) \subseteq \ker(\varphi) = 0$, hence $\iota$ is injective.

• What does "largest quotient of $N$" mean in the proof? (Being "largest" in what sense?)

By largest quotient of $N$, we mean that if $N/A$ is any other quotient of $N$, then $\ker \iota \subset A$. Note that any such $A$ arises as the kernel of some module homomorphism.

• How does one get "It follows that $N/\ker\iota$ is the unique largest quotient of $N$ that can be embedded in any $S$-module"?

At this point in the proof, we have already shown that if $\varphi:N\to L$ is any $R$-module homomorphism of $N$ into an $S$-module $L$, then $\ker\iota\subset\ker\varphi$ by using the universal property of the tensor product. In particular, if $\iota(n) = 0$, then $\varphi(n)= \Phi(\iota(n))=\Phi(0) = 0$.

Thus if $\varphi:N\to L$ is any $R$-module homomorphism of $N$ into an $S$-module, then there is an isomorphism theorem that tells us that $N/\ker\varphi$ can be embedded into $L$. Since $\iota$ is an $R$-module homomorphism, we can embed $N/\ker\iota$ into $S\otimes_R N$, and we have already shown it is the largest such quotient.

• Would anyone elaborate how "the last statement in the corollary follows immediately"?

If $N$ can be embedded as an $R$-submodule of some left $S$-module, then there is an $R$-module homomorphism $\varphi:N\to L$ such that $\ker\varphi = 0$. Since $N/\ker\iota$ is the largest quotient of $N$, $\ker\iota\subset 0$, hence $\ker\iota = 0$, so $\iota$ is injective.

If $\iota$ is injective, then $N/\ker\iota = N/0 \cong N$ can be embedded into $S\otimes_RN$.