[[Module theory MOC]]
# Tensor product of modules over a noncommutative ring

Unlike in the special case of the [[tensor product of modules over a commutative ring]], 
the general tensor product of modules may itself lack module structure.
Let $R$ be a (noncommutative) ring,
$M$ be a right $R$-[[module]] and $N$ be a left $R$-module.
The **tensor product** $M \otimes_{R} N$ is an [[abelian group]] such that the $R$-[[Balanced product|balanced]] maps from $M \times N$ are in correspondence with the [[Group homomorphism|group homomorphisms]] from $M \otimes_{R} N$, as defined by the [[#Universal property]]. 

## Universal property

Let $M$ be a right $R$-module and $N$ be a left $R$-module.
The **tensor product** is a pair consisting of an abelian group $M \otimes_{R} N$ together with an $R$-balanced map $(\otimes) : M \times N \to M \otimes_{R} N$ 
such that any $R$-balanced map $\varphi : M \times N \to G$ factorizes uniquely through $(\otimes)$ #m/def/module 

<p align="center"><img align="center" src="https://i.upmath.me/svg/%0A%5Cusetikzlibrary%7Bcalc%7D%0A%5Cusetikzlibrary%7Bdecorations.pathmorphing%7D%0A%5Ctikzset%7Bcurve%2F.style%3D%7Bsettings%3D%7B%231%7D%2Cto%20path%3D%7B(%5Ctikztostart)%0A%20%20%20%20..%20controls%20(%24(%5Ctikztostart)!%5Cpv%7Bpos%7D!(%5Ctikztotarget)!%5Cpv%7Bheight%7D!270%3A(%5Ctikztotarget)%24)%0A%20%20%20%20and%20(%24(%5Ctikztostart)!1-%5Cpv%7Bpos%7D!(%5Ctikztotarget)!%5Cpv%7Bheight%7D!270%3A(%5Ctikztotarget)%24)%0A%20%20%20%20..%20(%5Ctikztotarget)%5Ctikztonodes%7D%7D%2C%0A%20%20%20%20settings%2F.code%3D%7B%5Ctikzset%7Bquiver%2F.cd%2C%231%7D%0A%20%20%20%20%20%20%20%20%5Cdef%5Cpv%23%231%7B%5Cpgfkeysvalueof%7B%2Ftikz%2Fquiver%2F%23%231%7D%7D%7D%2C%0A%20%20%20%20quiver%2F.cd%2Cpos%2F.initial%3D0.35%2Cheight%2F.initial%3D0%7D%0A%25%20TikZ%20arrowhead%2Ftail%20styles.%0A%5Ctikzset%7Btail%20reversed%2F.code%3D%7B%5Cpgfsetarrowsstart%7Btikzcd%20to%7D%7D%7D%0A%5Ctikzset%7B2tail%2F.code%3D%7B%5Cpgfsetarrowsstart%7BImplies%5Breversed%5D%7D%7D%7D%0A%5Ctikzset%7B2tail%20reversed%2F.code%3D%7B%5Cpgfsetarrowsstart%7BImplies%7D%7D%7D%0A%25%20TikZ%20arrow%20styles.%0A%5Ctikzset%7Bno%20body%2F.style%3D%7B%2Ftikz%2Fdash%20pattern%3Don%200%20off%201mm%7D%7D%0A%25%20https%3A%2F%2Fq.uiver.app%2F%23q%3DWzAsMyxbMCwwLCJNIFxcdGltZXMgTiJdLFswLDIsIk0gXFxvdGltZXNfUiBOIl0sWzIsMCwiRyJdLFswLDEsIihcXG90aW1lcykiLDJdLFswLDIsIlxcdmFycGhpIl0sWzEsMiwiXFxleGlzdHMhXFxiYXJcXHZhcnBoaSIsMix7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ%3D%3D%0A%5Cbegin%7Btikzcd%7D%5Bampersand%20replacement%3D%5C%26%5D%0A%09%7BM%20%5Ctimes%20N%7D%20%5C%26%5C%26%20G%20%5C%5C%0A%09%5C%5C%0A%09%7BM%20%5Cotimes_R%20N%7D%0A%09%5Carrow%5B%22%5Cvarphi%22%2C%20from%3D1-1%2C%20to%3D1-3%5D%0A%09%5Carrow%5B%22%7B(%5Cotimes)%7D%22'%2C%20from%3D1-1%2C%20to%3D3-1%5D%0A%09%5Carrow%5B%22%7B%5Cexists!%5Cbar%5Cvarphi%7D%22'%2C%20dashed%2C%20from%3D3-1%2C%20to%3D1-3%5D%0A%5Cend%7Btikzcd%7D%0A#invert" alt="https://q.uiver.app/#q=WzAsMyxbMCwwLCJNIFxcdGltZXMgTiJdLFswLDIsIk0gXFxvdGltZXNfUiBOIl0sWzIsMCwiRyJdLFswLDEsIihcXG90aW1lcykiLDJdLFswLDIsIlxcdmFycGhpIl0sWzEsMiwiXFxleGlzdHMhXFxiYXJcXHZhcnBoaSIsMix7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==" /></p>

such that $\overline{\varphi}$ is a group homomorphism.

## Construction

Let $\mathbb{Z}^{(M \times N)}$ be a [[free module|free]] $\mathbb{Z}$-module [[free abelian group]] on $M \times N$ with the natural inclusion function $\iota : M \times N \hookrightarrow \mathbb{Z}^{(M \times N)}$.
Let $K$ denote the $\mathbb{Z}$-[[Submodule]] ([[subgroup]]) of $\mathbb{Z}^{(M \times N)}$ generated by any elements of the form
$$
\begin{align*}
\iota(m, n+n') - \iota(m, n) - \iota(m,n'); \\
\iota(m+m', n) - \iota(m,n) - \iota(m',n); \\
\iota(m \cdot r, n) - \iota(m,r \cdot n);
\end{align*}
$$
for any $m,m' \in M$, $n,n' \in N$, $r \in R$.
We construct the tensor product as the [[quotient module|quotient]] $\mathbb{Z}$-module
$$
\begin{align*}
M \otimes_{R} N = \mathbb{Z}^{(M \times N)} / K
\end{align*}
$$
with its natural projection $\pi: \mathbb{Z}^{(M \times N)}\twoheadrightarrow M \otimes_{R} N$,
so that the map
$$
\begin{align*}
(\otimes) = \pi \circ \iota : M \times N \to M \otimes_{R} N
\end{align*}
$$

> [!check]- Proof of the universal property
> By construction $(\otimes)$ is $R$-[[Balanced product|balanced]].
> Let $\varphi : M \times N \to G$ be $R$-balanced.
> By the [[Free module#Universal property|universal property of the free module]] we have a unique $\mathbb{Z}$-[[Module homomorphism|linear]] map $\tilde{\varphi}$ such that the following commutes:
> 
> <p align="center"><img align="center" src="https://i.upmath.me/svg/%0A%5Cusetikzlibrary%7Bcalc%7D%0A%5Cusetikzlibrary%7Bdecorations.pathmorphing%7D%0A%5Ctikzset%7Bcurve%2F.style%3D%7Bsettings%3D%7B%231%7D%2Cto%20path%3D%7B(%5Ctikztostart)%0A%20%20%20%20..%20controls%20(%24(%5Ctikztostart)!%5Cpv%7Bpos%7D!(%5Ctikztotarget)!%5Cpv%7Bheight%7D!270%3A(%5Ctikztotarget)%24)%0A%20%20%20%20and%20(%24(%5Ctikztostart)!1-%5Cpv%7Bpos%7D!(%5Ctikztotarget)!%5Cpv%7Bheight%7D!270%3A(%5Ctikztotarget)%24)%0A%20%20%20%20..%20(%5Ctikztotarget)%5Ctikztonodes%7D%7D%2C%0A%20%20%20%20settings%2F.code%3D%7B%5Ctikzset%7Bquiver%2F.cd%2C%231%7D%0A%20%20%20%20%20%20%20%20%5Cdef%5Cpv%23%231%7B%5Cpgfkeysvalueof%7B%2Ftikz%2Fquiver%2F%23%231%7D%7D%7D%2C%0A%20%20%20%20quiver%2F.cd%2Cpos%2F.initial%3D0.35%2Cheight%2F.initial%3D0%7D%0A%25%20TikZ%20arrowhead%2Ftail%20styles.%0A%5Ctikzset%7Btail%20reversed%2F.code%3D%7B%5Cpgfsetarrowsstart%7Btikzcd%20to%7D%7D%7D%0A%5Ctikzset%7B2tail%2F.code%3D%7B%5Cpgfsetarrowsstart%7BImplies%5Breversed%5D%7D%7D%7D%0A%5Ctikzset%7B2tail%20reversed%2F.code%3D%7B%5Cpgfsetarrowsstart%7BImplies%7D%7D%7D%0A%25%20TikZ%20arrow%20styles.%0A%5Ctikzset%7Bno%20body%2F.style%3D%7B%2Ftikz%2Fdash%20pattern%3Don%200%20off%201mm%7D%7D%0A%25%20https%3A%2F%2Fq.uiver.app%2F%23q%3DWzAsMyxbMCwwLCJNIFxcdGltZXMgTiJdLFswLDIsIlxcbWF0aGJiIFpbTSBcXHRpbWVzIE5dIl0sWzIsMCwiRyJdLFswLDEsIlxcaW90YSIsMl0sWzAsMiwiXFx2YXJwaGkiXSxbMSwyLCJcXGV4aXN0cyFcXHRpbGRlXFx2YXJwaGkiLDIseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0%3D%0A%5Cbegin%7Btikzcd%7D%5Bampersand%20replacement%3D%5C%26%5D%0A%09%7BM%20%5Ctimes%20N%7D%20%5C%26%5C%26%20G%20%5C%5C%0A%09%5C%5C%0A%09%7B%5Cmathbb%20Z%5BM%20%5Ctimes%20N%5D%7D%0A%09%5Carrow%5B%22%5Cvarphi%22%2C%20from%3D1-1%2C%20to%3D1-3%5D%0A%09%5Carrow%5B%22%5Ciota%22'%2C%20from%3D1-1%2C%20to%3D3-1%5D%0A%09%5Carrow%5B%22%7B%5Cexists!%5Ctilde%5Cvarphi%7D%22'%2C%20dashed%2C%20from%3D3-1%2C%20to%3D1-3%5D%0A%5Cend%7Btikzcd%7D%0A#invert" alt="https://q.uiver.app/#q=WzAsMyxbMCwwLCJNIFxcdGltZXMgTiJdLFswLDIsIlxcbWF0aGJiIFpbTSBcXHRpbWVzIE5dIl0sWzIsMCwiRyJdLFswLDEsIlxcaW90YSIsMl0sWzAsMiwiXFx2YXJwaGkiXSxbMSwyLCJcXGV4aXN0cyFcXHRpbGRlXFx2YXJwaGkiLDIseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0=" /></p>
> 
> and by the $R$-[[Balanced product|balance]] of $\varphi$ it follows $K \leq \ker \tilde{\varphi}$,
> so by the [[Quotient module#Universal property|universal property of the quotient module]] $\tilde{\varphi}$ factors uniquely through $\pi$,
> yielding the commutative diagram
> 
> <p align="center"><img align="center" src="https://i.upmath.me/svg/%0A%5Cusetikzlibrary%7Bcalc%7D%0A%5Cusetikzlibrary%7Bdecorations.pathmorphing%7D%0A%5Ctikzset%7Bcurve%2F.style%3D%7Bsettings%3D%7B%231%7D%2Cto%20path%3D%7B(%5Ctikztostart)%0A%20%20%20%20..%20controls%20(%24(%5Ctikztostart)!%5Cpv%7Bpos%7D!(%5Ctikztotarget)!%5Cpv%7Bheight%7D!270%3A(%5Ctikztotarget)%24)%0A%20%20%20%20and%20(%24(%5Ctikztostart)!1-%5Cpv%7Bpos%7D!(%5Ctikztotarget)!%5Cpv%7Bheight%7D!270%3A(%5Ctikztotarget)%24)%0A%20%20%20%20..%20(%5Ctikztotarget)%5Ctikztonodes%7D%7D%2C%0A%20%20%20%20settings%2F.code%3D%7B%5Ctikzset%7Bquiver%2F.cd%2C%231%7D%0A%20%20%20%20%20%20%20%20%5Cdef%5Cpv%23%231%7B%5Cpgfkeysvalueof%7B%2Ftikz%2Fquiver%2F%23%231%7D%7D%7D%2C%0A%20%20%20%20quiver%2F.cd%2Cpos%2F.initial%3D0.35%2Cheight%2F.initial%3D0%7D%0A%25%20TikZ%20arrowhead%2Ftail%20styles.%0A%5Ctikzset%7Btail%20reversed%2F.code%3D%7B%5Cpgfsetarrowsstart%7Btikzcd%20to%7D%7D%7D%0A%5Ctikzset%7B2tail%2F.code%3D%7B%5Cpgfsetarrowsstart%7BImplies%5Breversed%5D%7D%7D%7D%0A%5Ctikzset%7B2tail%20reversed%2F.code%3D%7B%5Cpgfsetarrowsstart%7BImplies%7D%7D%7D%0A%25%20TikZ%20arrow%20styles.%0A%5Ctikzset%7Bno%20body%2F.style%3D%7B%2Ftikz%2Fdash%20pattern%3Don%200%20off%201mm%7D%7D%0A%25%20https%3A%2F%2Fq.uiver.app%2F%23q%3DWzAsNCxbMCwwLCJNIFxcdGltZXMgTiJdLFswLDIsIlxcbWF0aGJiIFpbTSBcXHRpbWVzIE5dIl0sWzIsMCwiRyJdLFswLDQsIk0gXFxvdGltZXNfUiBOIl0sWzAsMSwiXFxpb3RhIiwyXSxbMCwyLCJcXHZhcnBoaSJdLFsxLDIsIlxcdGlsZGVcXHZhcnBoaSIsMl0sWzEsMywiXFxwaSIsMl0sWzMsMiwiXFxleGlzdHMhXFxiYXIgXFx2YXJwaGkiLDIseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbMCwzLCIoXFxvdGltZXMpIiwyLHsiY3VydmUiOjV9XV0%3D%0A%5Cbegin%7Btikzcd%7D%5Bampersand%20replacement%3D%5C%26%5D%0A%09%7BM%20%5Ctimes%20N%7D%20%5C%26%5C%26%20G%20%5C%5C%0A%09%5C%5C%0A%09%7B%5Cmathbb%20Z%5BM%20%5Ctimes%20N%5D%7D%20%5C%5C%0A%09%5C%5C%0A%09%7BM%20%5Cotimes_R%20N%7D%0A%09%5Carrow%5B%22%5Cvarphi%22%2C%20from%3D1-1%2C%20to%3D1-3%5D%0A%09%5Carrow%5B%22%5Ciota%22'%2C%20from%3D1-1%2C%20to%3D3-1%5D%0A%09%5Carrow%5B%22%7B(%5Cotimes)%7D%22'%2C%20curve%3D%7Bheight%3D30pt%7D%2C%20from%3D1-1%2C%20to%3D5-1%5D%0A%09%5Carrow%5B%22%7B%5Ctilde%5Cvarphi%7D%22'%2C%20from%3D3-1%2C%20to%3D1-3%5D%0A%09%5Carrow%5B%22%5Cpi%22'%2C%20from%3D3-1%2C%20to%3D5-1%5D%0A%09%5Carrow%5B%22%7B%5Cexists!%5Cbar%20%5Cvarphi%7D%22'%2C%20dashed%2C%20from%3D5-1%2C%20to%3D1-3%5D%0A%5Cend%7Btikzcd%7D%0A#invert" alt="https://q.uiver.app/#q=WzAsNCxbMCwwLCJNIFxcdGltZXMgTiJdLFswLDIsIlxcbWF0aGJiIFpbTSBcXHRpbWVzIE5dIl0sWzIsMCwiRyJdLFswLDQsIk0gXFxvdGltZXNfUiBOIl0sWzAsMSwiXFxpb3RhIiwyXSxbMCwyLCJcXHZhcnBoaSJdLFsxLDIsIlxcdGlsZGVcXHZhcnBoaSIsMl0sWzEsMywiXFxwaSIsMl0sWzMsMiwiXFxleGlzdHMhXFxiYXIgXFx2YXJwaGkiLDIseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbMCwzLCIoXFxvdGltZXMpIiwyLHsiY3VydmUiOjV9XV0=" /></p>
> 
> as required. <span class="QED"/>


## Tensor product of bimodules

Note that if $M$ is a $(T,R)$-[[bimodule]] and $R$ is a $(R,S)$-bimodule then $M \otimes_{R} N$ is naturally equipped with the structure of a $(T,S)$-bimodule.
If $R$ is commutative, then we recover the [[Tensor product of modules over a commutative ring]] by considering $R$-[[Bimodule|bimodules]] $M$ and $N$ this way.

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