B UCHBERGER T HEORY FOR E FFECTIVE A SSOCIATIVE R INGS T. M., Seven variations on standard bases , (1988) A solution if the ring is a vectorspace over a field A PEL J., Computational ideal theory in finitely generated extension rings , T.C.S. 224 (2000), 1–33 Extension to suitable rings which are algebra over a ring Gateva, Weispfenning and Passau group, Reinert, . . .
B UCHBERGER T HEORY FOR E FFECTIVE A SSOCIATIVE R INGS T. M., Seven variations on standard bases , (1988) A solution if the ring is a vectorspace over a field A PEL J., Computational ideal theory in finitely generated extension rings , T.C.S. 224 (2000), 1–33 Extension to suitable rings which are algebra over a ring Gateva, Weispfenning and Passau group, Reinert, . . .
B UCHBERGER T HEORY FOR E FFECTIVE A SSOCIATIVE R INGS T. M., Seven variations on standard bases , (1988) A solution if the ring is a vectorspace over a field A PEL J., Computational ideal theory in finitely generated extension rings , T.C.S. 224 (2000), 1–33 Extension to suitable rings which are algebra over a ring Gateva, Weispfenning and Passau group, Reinert, . . .
B UCHBERGER T HEORY FOR E FFECTIVE A SSOCIATIVE R INGS T. M., Seven variations on standard bases , (1988) A solution if the ring is a vectorspace over a field A PEL J., Computational ideal theory in finitely generated extension rings , T.C.S. 224 (2000), 1–33 Extension to suitable rings which are algebra over a ring Gateva, Weispfenning and Passau group, Reinert, . . .
T HEOREM For an (associative but not necessarily commutative) ring with identity A , there is a (not necessarily finite nor necessarily countable) set Z and a projection Π : Q := Z � Z � ։ A so that, denoting I ⊂ Q = Z � Z � the bilateral ideal I := ker(Π) , we have A = Q / I .
E FFECTIVELY GIVEN A SSOCIATIVE R INGS Let R be an (associative but not necessarily commutative) ring with identity 1 R and A another (associative but not necessarily commutative) ring with identity 1 A which is a left module on R . We consider A to be effectively given when we are given • sets v := { x 1 , . . . , x j , . . . } , V := { X 1 , . . . , X i , . . . } , which are countable and • Z := v ⊔ V = { x 1 , . . . , x j , . . . , X 1 , . . . , X i , . . . } ; • rings R := Z � v � ⊂ Q := Z � Z � ; • projections π : R = Z � x 1 , . . . , x j , . . . � ։ R and • Π : Q := Z � x 1 , . . . , x j , . . . , X 1 , . . . , X i , . . . � ։ A which satisfies Π( x j ) = π ( x j ) 1 A , for each x j ∈ v , so that Π ( R ) = { r 1 A : r ∈ R } ⊂ A .
E FFECTIVELY GIVEN A SSOCIATIVE R INGS Let R be an (associative but not necessarily commutative) ring with identity 1 R and A another (associative but not necessarily commutative) ring with identity 1 A which is a left module on R . We consider A to be effectively given when we are given • sets v := { x 1 , . . . , x j , . . . } , V := { X 1 , . . . , X i , . . . } , which are countable and • Z := v ⊔ V = { x 1 , . . . , x j , . . . , X 1 , . . . , X i , . . . } ; • rings R := Z � v � ⊂ Q := Z � Z � ; • projections π : R = Z � x 1 , . . . , x j , . . . � ։ R and • Π : Q := Z � x 1 , . . . , x j , . . . , X 1 , . . . , X i , . . . � ։ A which satisfies Π( x j ) = π ( x j ) 1 A , for each x j ∈ v , so that Π ( R ) = { r 1 A : r ∈ R } ⊂ A .
E FFECTIVELY GIVEN A SSOCIATIVE R INGS Let R be an (associative but not necessarily commutative) ring with identity 1 R and A another (associative but not necessarily commutative) ring with identity 1 A which is a left module on R . We consider A to be effectively given when we are given • sets v := { x 1 , . . . , x j , . . . } , V := { X 1 , . . . , X i , . . . } , which are countable and • Z := v ⊔ V = { x 1 , . . . , x j , . . . , X 1 , . . . , X i , . . . } ; • rings R := Z � v � ⊂ Q := Z � Z � ; • projections π : R = Z � x 1 , . . . , x j , . . . � ։ R and • Π : Q := Z � x 1 , . . . , x j , . . . , X 1 , . . . , X i , . . . � ։ A which satisfies Π( x j ) = π ( x j ) 1 A , for each x j ∈ v , so that Π ( R ) = { r 1 A : r ∈ R } ⊂ A .
E FFECTIVELY GIVEN A SSOCIATIVE R INGS Let R be an (associative but not necessarily commutative) ring with identity 1 R and A another (associative but not necessarily commutative) ring with identity 1 A which is a left module on R . We consider A to be effectively given when we are given • sets v := { x 1 , . . . , x j , . . . } , V := { X 1 , . . . , X i , . . . } , which are countable and • Z := v ⊔ V = { x 1 , . . . , x j , . . . , X 1 , . . . , X i , . . . } ; • rings R := Z � v � ⊂ Q := Z � Z � ; • projections π : R = Z � x 1 , . . . , x j , . . . � ։ R and • Π : Q := Z � x 1 , . . . , x j , . . . , X 1 , . . . , X i , . . . � ։ A which satisfies Π( x j ) = π ( x j ) 1 A , for each x j ∈ v , so that Π ( R ) = { r 1 A : r ∈ R } ⊂ A .
E FFECTIVELY GIVEN A SSOCIATIVE R INGS Π : Q := Z � x 1 , . . . , x j , . . . , X 1 , . . . , X i , . . . � ։ A π : R = Z � x 1 , . . . , x j , . . . � ։ R Thus denoting • I := ker(Π) ⊂ Q and • I := I ∩ R = ker( π ) ⊂ R , we have A = Q / I and R = R / I ; moreover we can wlog assume that R ⊂ A . Q as Z -module: using as alphabet V all symbols representing the primes. Further, when considering A as effectively given in this way, we explicitly require that i � X i x j ≡ π ( a lij ) X l + π ( a 0 ij ) mod I , a lij ∈ Z � v � , ∀ X i ∈ V , x j ∈ v . l =1 If not Z � X , Y � as left Z [ X ]-module requires 1 ≥ X ≥ X 2
E FFECTIVELY GIVEN A SSOCIATIVE R INGS Π : Q := Z � x 1 , . . . , x j , . . . , X 1 , . . . , X i , . . . � ։ A π : R = Z � x 1 , . . . , x j , . . . � ։ R Thus denoting • I := ker(Π) ⊂ Q and • I := I ∩ R = ker( π ) ⊂ R , we have A = Q / I and R = R / I ; moreover we can wlog assume that R ⊂ A . Q as Z -module: using as alphabet V all symbols representing the primes. Further, when considering A as effectively given in this way, we explicitly require that i � X i x j ≡ π ( a lij ) X l + π ( a 0 ij ) mod I , a lij ∈ Z � v � , ∀ X i ∈ V , x j ∈ v . l =1 If not Z � X , Y � as left Z [ X ]-module requires 1 ≥ X ≥ X 2
E FFECTIVELY GIVEN A SSOCIATIVE R INGS Π : Q := Z � x 1 , . . . , x j , . . . , X 1 , . . . , X i , . . . � ։ A π : R = Z � x 1 , . . . , x j , . . . � ։ R Thus denoting • I := ker(Π) ⊂ Q and • I := I ∩ R = ker( π ) ⊂ R , we have A = Q / I and R = R / I ; moreover we can wlog assume that R ⊂ A . Q as Z -module: using as alphabet V all symbols representing the primes. Further, when considering A as effectively given in this way, we explicitly require that i � X i x j ≡ π ( a lij ) X l + π ( a 0 ij ) mod I , a lij ∈ Z � v � , ∀ X i ∈ V , x j ∈ v . l =1 If not Z � X , Y � as left Z [ X ]-module requires 1 ≥ X ≥ X 2
E FFECTIVELY GIVEN A SSOCIATIVE R INGS Π : Q := Z � x 1 , . . . , x j , . . . , X 1 , . . . , X i , . . . � ։ A π : R = Z � x 1 , . . . , x j , . . . � ։ R Thus denoting • I := ker(Π) ⊂ Q and • I := I ∩ R = ker( π ) ⊂ R , we have A = Q / I and R = R / I ; moreover we can wlog assume that R ⊂ A . Q as Z -module: using as alphabet V all symbols representing the primes. Further, when considering A as effectively given in this way, we explicitly require that i � X i x j ≡ π ( a lij ) X l + π ( a 0 ij ) mod I , a lij ∈ Z � v � , ∀ X i ∈ V , x j ∈ v . l =1 If not Z � X , Y � as left Z [ X ]-module requires 1 ≥ X ≥ X 2
E FFECTIVELY GIVEN A SSOCIATIVE R INGS Π : Q := Z � x 1 , . . . , x j , . . . , X 1 , . . . , X i , . . . � ։ A = Q / I π : R = Z � x 1 , . . . , x j , . . . � ։ R = R / I If we fix a term-ordering < on � Z � we can assume I to be given via its Gr¨ obner basis G w.r.t. < and, if < satisfies X i > t for each t ∈ � v � and X i ∈ V also I is given via its Gr¨ obner basis G 0 := G ∩ R w.r.t. < .
E FFECTIVELY GIVEN A SSOCIATIVE R INGS Π : Q := Z � x 1 , . . . , x j , . . . , X 1 , . . . , X i , . . . � ։ A = Q / I π : R = Z � x 1 , . . . , x j , . . . � ։ R = R / I If we fix a term-ordering < on � Z � we can assume I to be given via its Gr¨ obner basis G w.r.t. < and, if < satisfies X i > t for each t ∈ � v � and X i ∈ V also I is given via its Gr¨ obner basis G 0 := G ∩ R w.r.t. < .
E FFECTIVELY GIVEN A SSOCIATIVE R INGS Π : Q := Z � x 1 , . . . , x j , . . . , X 1 , . . . , X i , . . . � ։ A = Q / I π : R = Z � x 1 , . . . , x j , . . . � ։ R = R / I If we fix a term-ordering < on � Z � we can assume I to be given via its Gr¨ obner basis G w.r.t. < and, if < satisfies X i > t for each t ∈ � v � and X i ∈ V also I is given via its Gr¨ obner basis G 0 := G ∩ R w.r.t. < .
E FFECTIVELY GIVEN A SSOCIATIVE R INGS Π : Q := Z � x 1 , . . . , x j , . . . , X 1 , . . . , X i , . . . � ։ A = Q / I π : R = Z � x 1 , . . . , x j , . . . � ։ R = R / I If we fix a term-ordering < on � Z � we can assume I to be given via its Gr¨ obner basis G w.r.t. < and, if < satisfies X i > t for each t ∈ � v � and X i ∈ V also I is given via its Gr¨ obner basis G 0 := G ∩ R w.r.t. < . i � X i x j ≡ π ( a lij ) X l + π ( a 0 ij ) mod I , a lij ∈ Z � v � , ∀ X i ∈ V , x j ∈ v l =1
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