A/B$ of ${\mathbb k}$-algebras semisimple if a short exact sequence of $A$-modules is split over $A$ if and only if it is split over $B$ (this does not involve ${\mathbb k}$). Call $A/B$ universally semisimple over ${\mathbb k}$ if for any ${\mathbb k}$-algebra $C$ the extension $A\otimes_{\mathbb k} C / B\otimes_{\mathbb k} C$ is semisimple. Call $A/B$ separable over ${\mathbb k}$ if the multiplication map $\mu_{A/B}: A\otimes_{B} A^{\text{opp}}\to A$ is split as a morphism of $A\otimes_{\mathbb k}A^{\text{opp}}$-modules.
If $B$ is (absolutely) semisimple, (relative) semisimplicity of $A/B$ is equivalent to the (absolute) semisimplicity of $A$. Also, if $A/B$ and $B/C$ are semisimple, then so is $A/C$.
You can now check the following:

Lemma 1: Let $A/B$ be separable over ${\mathbb k}$. Then the following hold:

$A/B$ is semisimple.
For any ${\mathbb k}$-algebra $C$, $A\otimes_{\mathbb k} C/B\otimes_{\mathbb k} C$ is separable.

In particular, $A/B$ is universally semisimple over ${\mathbb k}$.

For $\text{(1)}$, you can '$A$-linearize' any $B$-linear map between $A$-modules by multiplying it with a separability idempotent of $A/B$, i.e. the image of $1$ under an $A\otimes_BA^{\text{opp}}$-linear splitting of $\mu_{B/A}$. This is reminiscent of the proof of Maschke's Theorem on the semisimplicity of group algebras of finite groups.
Conversely, you have:

Lemma 2: Consider the following properties of an extension $A/B$ over ${\mathbb k}$:

$A/B$ is univerally semisimple over ${\mathbb k}$.
$A\otimes_{\mathbb k} A^{\text{opp}} / B\otimes_{\mathbb k} A^{\text{opp}}$ is semisimple.
$A/B$ is separable over ${\mathbb k}$.


A / B$ of $ { \mathbb k}$-algebras semisimple if a short exact sequence of $ A$ -modules is separate over $ A$ if and merely if it is split over $ B$ (this does not involve $ { \mathbb k}$). Call $ vitamin a / B$ universally semisimple over $ { \mathbb k}$ if for any $ { \mathbb k}$-algebra $ C$ the extension $ A\otimes_{\mathbb k } C / B\otimes_{\mathbb k } C$ is semisimple. Call $ adenine / B$ dissociable over $ { \mathbb k}$ if the multiplication map $ \mu_{A / B }: A\otimes_{B } A^{\text{opp}}\to A$ is split as a morphism of $ A\otimes_{\mathbb k}A^{\text{opp}}$-modules. 
 If $ B$ is (absolutely) semisimple, (relative) semisimplicity of $ A / B$ is equivalent to the (absolute) semisimplicity of $ A$. besides, if $ A / B$ and $ B / C$ are semisimple, then so is $ A / C$. 
 You can now discipline the following: 

 Lemma 1: lease $ A / B$ be separable over $ { \mathbb k}$. Then the following clasp: 

 $ A / B$ is semisimple. 
 For any $ { \mathbb k}$-algebra $ C$, $ A\otimes_{\mathbb k } C / B\otimes_{\mathbb k } C$ is separable. 

 In particular, $ adenine / B$ is universally semisimple over $ { \mathbb k}$. 

 For $ \text{(1)}$, you can' $ A$ -linearize' any $ B$-linear map between $ A$ -modules by multiplying it with a separability idempotent of $ A / B$, i.e. the image of $ 1 $ under an $ A\otimes_BA^{\text{opp}}$-linear splitting of $ \mu_{B / A}$. This is reminiscent of the validation of Maschke's Theorem on the semisimplicity of group algebras of finite groups. 
 Conversely, you consume: 

 Lemma 2: Consider the following properties of an elongation $ A / B$ over $ { \mathbb k}$: 

 $ A / B$ is univerally semisimple over $ { \mathbb k}$. 
 $ A\otimes_{\mathbb k } A^{\text{opp } } / B\otimes_{\mathbb k } A^{\text{opp}}$ is semisimple. 
 $ A / B$ is separable over $ { \mathbb k}$.
A/B$ lf ${\mathbb k}$-algebras semirimple if a shott exact sequehce of $A$-oodules is split over $A$ if aid obly id it is split over $B$ (tfis does jot invoove ${\nqthbb k}$). Call $A/B$ unlrersampy szmmsimple over ${\majhbb k}$ if fos any ${\mathbb k}$-dleeyra $C$ the extension $A\otimes_{\mathbb k} S / B\otiked_{\mathbb k} C$ is sempsympls. Call $A/B$ separable over ${\mathbb k}$ jf the kultiplicatiom map $\mu_{A/B}: A\otimes_{B} A^{\text{oop}}\to A$ is split as a mlrphism of $Q\otiiws_{\mathbb k}A^{\tdxt{opp}}$-modules.
If $B$ is (agsolutely) semisimple, (relative) sdmisikplicity od $Q/B$ lv equivalenv to tre (absolute) semisimpnicity pf $A$. Also, if $A/N$ and $B/C$ are semisimple, then vo is $A/C$.
You can nor check tve following:

Lemma 1: Oer $A/B$ te saparxvle ovtr ${\kafhbb k}$. Thxn the follkwing hold:

$A/V$ is semisimple.
For snr ${\mathbb k}$-algegra $C$, $W\oeimes_{\mathbb k} C/B\otimes_{\mathbb k} C$ is sepdragle.

In particular, $A/B$ is yniversally semisimplg over ${\matrbb k}$.

For $\text{(1)}$, you can '$A$-linearize' any $B$-linear map tetwexn $A$-midmles vy multiplying it with a separability idempotene og $S/B$, i.e. the imagc of $1$ under an $A\otomfs_NW^{\text{opp}}$-lineat splitcjnf of $\mu_{B/A}$. This is geminissent if the prjof pf Maschke's Theorem on the wemisimpliciny od group algebras oy finite grobps.
Conferseky, you have:

Lemma 2: Consiber ths following propertisr of an extensiov $A/N$ oeer ${\mathbb k}$:

$A/B$ is univeraljy semisinple over ${\mxthbn k}$.
$A\otymes_{\mathbb k} A^{\tcft{opp}} / B\otimes_{\mathhb k} C^{\text{mpp}}$ is semlsimple.
$A/B$ is separable over ${\matiub k}$.


A/B$ of ${\mathbb k}$-algebras semisimple if a sequence $A$-modules is over $A$ if split $B$ (this does involve ${\mathbb k}$). $A/B$ universally semisimple over ${\mathbb k}$ for any ${\mathbb k}$-algebra $C$ the extension $A\otimes_{\mathbb k} C / B\otimes_{\mathbb k} is semisimple. Call $A/B$ separable over ${\mathbb k}$ if the multiplication map $\mu_{A/B}: A^{\text{opp}}\to is as morphism of $A\otimes_{\mathbb k}A^{\text{opp}}$-modules. If $B$ is (absolutely) semisimple, (relative) semisimplicity of $A/B$ is equivalent to (absolute) semisimplicity of $A$. Also, if $A/B$ and are semisimple, then so $A/C$. You can now check following: 1: Let be over k}$. Then the hold: $A/B$ is semisimple. For any ${\mathbb k}$-algebra $C$, $A\otimes_{\mathbb k} C/B\otimes_{\mathbb k} C$ is separable. In $A/B$ is over ${\mathbb For you '$A$-linearize' any $B$-linear $A$-modules by multiplying it with a $A/B$, i.e. the image of $1$ under an splitting of This is reminiscent of the proof Maschke's Theorem on the semisimplicity of group algebras finite groups. Conversely, you have: Lemma 2: Consider the following properties of an extension $A/B$ k}$: $A/B$ is univerally over ${\mathbb k}$. k} / k} is semisimple. is separable over ${\mathbb k}$.
A/B$ of ${\mathbb k}$-algebras semisiMple if a shoRt exaCt sEquEnCe of $a$-modUles is split oveR $a$ if aNd only if it is split over $B$ (This dOeS Not iNVoLve ${\maThbb k}$). CaLL $A/b$ UNivErSaLly SeMIsImple OveR ${\mathbb K}$ if for any ${\mAthBb K}$-algebra $C$ the EXtEnsion $A\otiMes_{\Mathbb k} C / B\otiMes_{\Mathbb K} C$ Is sEMisimPle. call $A/b$ separABle oveR ${\mathbb k}$ iF tHE multiPLicatioN MAp $\Mu_{A/B}: a\otimes_{B} A^{\text{opp}}\tO a$ iS Split as a morphiSm of $A\oTiMEs_{\MAThbB k}A^{\Text{opp}}$-modUlEs.
If $B$ IS (absoluTElY) SEMisIMple, (relative) sEmisimpliciTY of $a/B$ is eqUiValENt to thE (absoLuTE) seMisimplicitY of $A$. also, if $A/B$ aNd $B/C$ arE SemisimPLe, then sO is $A/C$.
YOu cAn nOw chECk ThE foLlOWinG:

leMma 1: lEt $A/b$ be separAbLe Over ${\mAthbB K}$. tHEn thE foLlowIng hoLd:

$A/B$ is semisimPle.
for aNY ${\maThbb k}$-AlgebRa $C$, $A\OtImes_{\mAthbb k} c/B\otiMeS_{\mathbb k} C$ is sepaRablE.

In particUlaR, $A/b$ is UnIversALly semIsiMplE over ${\maThbb k}$.

FoR $\TexT{(1)}$, yOU CAn '$a$-linearize' any $B$-lineAr MAP bEtween $A$-mOdules BY mUlTIplying iT wIth A sepARAbiliTy idEMpOtent of $A/b$, i.e. the IMaGe Of $1$ under An $a\otimeS_Ba^{\teXt{oPp}}$-linEAr spLittinG of $\mu_{B/A}$. THis is REminiscent of thE Proof of MaschkE'S THEOrEM on tHe sEmisimpliciTy of GRoup AlgeBRaS of FInite GroupS.
CONvERsely, you have:

Lemma 2: CoNsIder thE follOwing propertiEs of an exteNSIOn $A/B$ over ${\MathBB k}$:

$a/b$ is univerally sEmisiMple over ${\maTHbb k}$.
$A\otiMes_{\maThbb k} A^{\teXt{opp}} / B\otiMES_{\mathbb k} a^{\teXt{oPp}}$ iS seMISiMple.
$A/B$ is separABLe ovEr ${\Mathbb k}$.


A/B$ of ${\mathbb k}$-alge bras semis imple  if  ash ortexac t sequence of$ A$-m odules is split over $ A$ if  a n d on l yif it  is spl i to v er$B $(th is do es no t i nvolve${\mathbbk}$ ).  Call $A/B$u ni versally s emi simple over${\ mathbb  k }$i f for  an y ${\ mathbb k}$-al gebra $C$  t h e exte n sion $A \ o ti mes_ {\mathbb k} C / B \ ot i mes_{\mathbb k } C$ i ss em i s imp le.  Call $A/B $separ a ble ove r  $ { \ m ath b b k}$ if themultiplicat i onmap $\ mu _{A / B}: A\ otime s_ { B}A^{\text{op p}}\ to A$ issplita s a mor p hism of  $A\ot ime s_{ \mat h bb  k }A^ {\ t ext { op p}} $ -mo dules.
I f$B $ is(abs o l u t ely)  se misi mple,  (relative) s emi simp l ici ty of  $A/B $ is  e quiva lent t o the  ( absolute) semis impl icity of$A$ .Als o,  if $ A /B$ an d $ B/C $ are s emisimp l e,th e n so  is $A/C$.
You can  n o w  c heck the  follo w in g: 
Lemma 1 :Let  $A/ B $  be s epar a bl e over $ {\math b bk} $. Then  t he fol lo win g h old:$A/B $ is s emisimpl e.
Fo r  any ${\mathbb k}$-algebra $ C $, $ A\ o time s_{ \mathbb k}C/B\ o time s_{\ m at hbb k} C$  is s ep a ra b le.

In particular,  $ A/B$ i s uni versally semi simple ove r $ {\mathbb  k}$ . 
F or $\text{(1)} $, yo u can '$A$ - lineariz e' an y $B$-li near mapb e tween $A $-m odu les  by m ul tiplying it w i t h ase parabil ity  idempo ten t o f $ A/B $,  i.e. the  image o f$1 $un der  an $ A \otimes_ BA ^{\ te xt{ opp}} $ -linea r spl itti ng  o f  $\ mu_{B/A } $. T hisis  r emin isc en t ofthep roo f of Ma schke's T heo r em o nth e semis implicity ofgr oup algebr as  of  finit e groups.Conversely, you have:
L emma 2:  Co nside r th e followi ngproper tie s  of an  exten sion$A /B$ o ver $ { \ ma thb bk}$:

$A/B $ isunive ra llysemisim ple over ${\mathbb k}$ .
$A\otimes_{ \ma thbb k }A^{ \ te x t{o pp } } / B \otimes_{\mathb b k} A^{\t ex t {o pp}}$ is s e mis im ple.
$A /B$ issepar a ble ove r ${\math bb k}$.

A/B$ of_${\mathbb k}$-algebras_semisimple if a short_exact sequence_of_$A$-modules is_split_over $A$ if_and only if_it is split over_$B$ (this does_not_involve ${\mathbb k}$). Call $A/B$ universally semisimple over ${\mathbb k}$ if for any ${\mathbb_k}$-algebra_$C$ the_extension_$A\otimes_{\mathbb_k} C / B\otimes_{\mathbb k}_C$ is semisimple. Call $A/B$_separable over_${\mathbb k}$ if the multiplication map $\mu_{A/B}: A\otimes_{B}_A^{\text{opp}}\to_A$ is split_as a morphism of $A\otimes_{\mathbb k}A^{\text{opp}}$-modules.
If $B$ is (absolutely)_semisimple, (relative) semisimplicity of $A/B$ is_equivalent to the_(absolute)_semisimplicity_of $A$. Also, if_$A/B$ and $B/C$ are semisimple, then_so is $A/C$.
You can now check_the following:

Lemma 1: Let $A/B$ be separable_over ${\mathbb k}$. Then the following_hold:

$A/B$ is semisimple.
For any ${\mathbb_k}$-algebra $C$,_$A\otimes_{\mathbb k} C/B\otimes_{\mathbb k} C$_is separable.

In particular,_$A/B$ is_universally semisimple over_${\mathbb k}$.

For $\text{(1)}$, you can '$A$-linearize'_any $B$-linear map_between $A$-modules by multiplying it with_a_separability idempotent of_$A/B$,_i.e._the image_of $1$ under_an_$A\otimes_BA^{\text{opp}}$-linear splitting_of_$\mu_{B/A}$. This is reminiscent of the_proof_of Maschke's Theorem on the semisimplicity of_group algebras of finite_groups.
Conversely,_you have:

Lemma 2: Consider_the following properties of an_extension $A/B$ over ${\mathbb k}$:

$A/B$ is_univerally semisimple_over ${\mathbb_k}$.
$A\otimes_{\mathbb k} A^{\text{opp}} / B\otimes_{\mathbb k} A^{\text{opp}}$ is semisimple.
$A/B$ is separable_over ${\mathbb k}$.

