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Scalars are commutative after being lifted into an associative algebra (#5015)
* [add] assa2ass2; [shorten] assa2ass * [add] asclcom * [add] elmgpcntrd * [prove] asclcntr; [add] asclelbas * [minimize] asclcom * [delete] assascacom and assascacrng * [improve] explanation of asclcom * [rename] structure map of associative algebra might be non-injective; [fix] algebraic scalars map -> algebra scalars map
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changes-set.txt

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@@ -92,6 +92,7 @@ make a github issue.)
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DONE:
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Date Old New Notes
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11-Sep-25 cascl [same] revised - algSc may be non-injective
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10-Sep-25 soeq12d [same] Moved from SO's mathbox to main set.mm
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10-Sep-25 freq12d [same] Moved from SO's mathbox to main set.mm
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10-Sep-25 weeq12d [same] Moved from SO's mathbox to main set.mm

set.mm

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@@ -283964,7 +283964,7 @@ the same dimension over the same (nonzero) ring. (Contributed by AV,
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$( Algebraic span function. $)
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casp $a class AlgSpan $.
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$( Class of algebra scalar injection function. $)
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$( Class of algebra scalars function. $)
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cascl $a class algSc $.
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${
@@ -284376,7 +284376,7 @@ the same dimension over the same (nonzero) ring. (Contributed by AV,
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asclfval.k $e |- K = ( Base ` F ) $.
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asclfval.s $e |- .x. = ( .s ` W ) $.
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asclfval.o $e |- .1. = ( 1r ` W ) $.
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$( Function value of the algebraic scalars function. (Contributed by Mario
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$( Function value of the algebra scalars function. (Contributed by Mario
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Carneiro, 8-Mar-2015.) $)
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asclfval $p |- A = ( x e. K |-> ( x .x. .1. ) ) $=
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( vw cascl cfv cmpt cbs csca eqtr4di c0 cv cvv wcel wceq cur cvsca fveq2d
@@ -284533,8 +284533,8 @@ the same dimension over the same (nonzero) ring. (Contributed by AV,
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$d x y A $. $d x y F $. $d x y W $.
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asclrhm.a $e |- A = ( algSc ` W ) $.
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asclrhm.f $e |- F = ( Scalar ` W ) $.
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$( The scalar injection is a ring homomorphism. (Contributed by Mario
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Carneiro, 8-Mar-2015.) $)
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$( The algebra scalars function is a ring homomorphism. (Contributed by
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Mario Carneiro, 8-Mar-2015.) $)
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asclrhm $p |- ( W e. AssAlg -> A e. ( F RingHom W ) ) $=
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( vx vy casa wcel cbs cfv cmulr cur assasca assaring assalmod ascl1 cv co
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eqid wceq ascldimul 3expb asclghm isrhm2d ) CHIZFGBJKZBCBLKZCLKZBMKZACMKZ
@@ -284547,7 +284547,7 @@ the same dimension over the same (nonzero) ring. (Contributed by AV,
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rnascl.a $e |- A = ( algSc ` W ) $.
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rnascl.o $e |- .1. = ( 1r ` W ) $.
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rnascl.n $e |- N = ( LSpan ` W ) $.
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$( The set of injected scalars is also interpretable as the span of the
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$( The set of lifted scalars is also interpretable as the span of the
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identity. (Contributed by Mario Carneiro, 9-Mar-2015.) $)
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rnascl $p |- ( W e. AssAlg -> ran A = ( N ` { .1. } ) ) $=
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( vx vy casa wcel crn cv cvsca cfv co wceq csca cbs eqid cab csn asclfval
@@ -284620,7 +284620,7 @@ the same dimension over the same (nonzero) ring. (Contributed by AV,
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$d S x $. $d W x $. $d X x $.
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ressascl.a $e |- A = ( algSc ` W ) $.
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ressascl.x $e |- X = ( W |`s S ) $.
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$( The injection of scalars is invariant between subalgebras and
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$( The lifting of scalars is invariant between subalgebras and
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superalgebras. (Contributed by Mario Carneiro, 9-Mar-2015.) $)
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ressascl $p |- ( S e. ( SubRing ` W ) -> A = ( algSc ` X ) ) $=
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( vx csubrg cfv wcel csca cbs cv cur cvsca co cmpt cascl eqid asclfval
@@ -835145,6 +835145,107 @@ have GLB (expanded version). (Contributed by Zhi Wang,
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$}
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$(
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=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
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Rings
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=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
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$)
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$(
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-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
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Multiplicative Group
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-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
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$)
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${
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$d B y $. $d M y $. $d R y $. $d X y $. $d ph y $.
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elmgpcntrd.b $e |- B = ( Base ` R ) $.
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elmgpcntrd.m $e |- M = ( mulGrp ` R ) $.
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elmgpcntrd.z $e |- Z = ( Cntr ` M ) $.
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elmgpcntrd.x $e |- ( ph -> X e. B ) $.
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elmgpcntrd.y $e |- ( ( ph /\ y e. B ) ->
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( X ( .r ` R ) y ) = ( y ( .r ` R ) X ) ) $.
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$( The center of a ring. (Contributed by Zhi Wang, 11-Sep-2025.) $)
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elmgpcntrd $p |- ( ph -> X e. Z ) $=
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( wcel cv cmulr cfv co wceq wral ralrimiva mgpbas eqid mgpplusg sylanbrc
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elcntr ) AFCMFBNZDOPZQUFFUGQRZBCSFGMKAUHBCLTBFCUGEGCDEIHUADUGEIUGUBUCJUEU
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D $.
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$}
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$(
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=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
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Associative algebras
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=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
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$)
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$(
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-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
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Definition and basic properties
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-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
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$)
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${
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asclelbas.a $e |- A = ( algSc ` W ) $.
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asclelbas.f $e |- F = ( Scalar ` W ) $.
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asclelbas.b $e |- B = ( Base ` F ) $.
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asclelbas.w $e |- ( ph -> W e. AssAlg ) $.
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asclelbas.c $e |- ( ph -> C e. B ) $.
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$( Lifted scalars are in the base set of the algebra. (Contributed by Zhi
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Wang, 11-Sep-2025.) $)
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asclelbas $p |- ( ph -> ( A ` C ) e. ( Base ` W ) ) $=
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( cfv cur cvsca co cbs wcel wceq eqid syl asclval casa clmod assalmod crg
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assaring ringidcl 3syl lmodvscld eqeltrd ) ADBLZDFMLZFNLZOZFPLZADCQUKUNRK
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BUMULECFDGHIUMSZULSZUATADUMECUOFULUOSZHUPIAFUBQZFUCQJFUDTKAUSFUEQULUOQJFU
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FUOFULURUQUGUHUIUJ $.
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${
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$d A x $. $d C x $. $d M x $. $d W x $. $d ph x $.
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asclcntr.m $e |- M = ( mulGrp ` W ) $.
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$( The algebra scalars function maps into the center of the algebra.
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Equivalently, a lifted scalar is a center of the algebra.
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(Contributed by Zhi Wang, 11-Sep-2025.) $)
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asclcntr $p |- ( ph -> ( A ` C ) e. ( Cntr ` M ) ) $=
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( vx cbs cfv eqid wcel co adantr ccntr asclelbas cv wa casa cmulr simpr
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wceq w3a cvsca asclmul1 asclmul2 eqtr4d syl3anc elmgpcntrd ) ANGOPZGFDB
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PZFUAPZUPQZMURQABCDEGHIJKLUBANUCZUPRZUDGUERZDCRZVAUQUTGUFPZSZUTUQVDSZUH
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AVBVAKTAVCVALTAVAUGVBVCVAUIVEDUTGUJPZSVFBDVGVDECUPGUTHIJUSVDQZVGQZUKBDV
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GVDECUPGUTHIJUSVHVIULUMUNUO $.
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$}
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${
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asclcom.m $e |- .* = ( .r ` F ) $.
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asclcom.d $e |- ( ph -> D e. B ) $.
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$( Scalars are commutative after being lifted.
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However, the scalars themselves are not necessarily commutative if the
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algebra is not a faithful module. For example, Let ` F ` be the 2 by
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2 upper triangular matrix algebra over a commutative ring ` W ` . It
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is provable that ` F ` is in general non-commutative. Define scalar
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multiplication ` C .x. X ` as multipying the top-left entry, which is
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a "vector" element of ` W ` , of the "scalar" ` C ` , which is now an
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upper triangular matrix, with the "vector" ` X e. ( Base `` W ) ` .
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Equivalently, the algebra scalars function is not necessarily
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injective unless the algebra is faithful. Therefore, all "scalar
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injection" was renamed.
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Alternate proof involves ~ assa2ass , ~ assa2ass2 , and ~ asclval , by
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setting ` X ` and ` Y ` the multiplicative identity of the algebra.
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(Contributed by Zhi Wang, 11-Sep-2025.) $)
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asclcom $p |- ( ph -> ( A ` ( C .* D ) ) = ( A ` ( D .* C ) ) ) $=
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( cfv co wcel wceq eqid cmulr cbs asclelbas w3a cvsca asclmul1 asclmul2
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casa eqtr4d syl3anc ascldimul 3eqtr4d ) ADBPZEBPZHUAPZQZUNUMUOQZDEGQBPZ
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EDGQBPZAHUHRZDCRZUNHUBPZRZUPUQSLMABCEFHIJKLOUCUTVAVCUDUPDUNHUEPZQUQBDVD
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UOFCVBHUNIJKVBTZUOTZVDTZUFBDVDUOFCVBHUNIJKVEVFVGUGUIUJAUTVAECRZURUPSLMO
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BDEGUOFCHIJKVFNUKUJAUTVHVAUSUQSLOMBEDGUOFCHIJKVFNUKUJUL $.
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$}
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$}
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$(
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=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
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