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### Page 1401 of Mathematical Reviews Vol. , Issue 85d [page]

1985 Mathematical Reviews
There are three main results in this paper, all concerning the enveloping algebra A of a solvable Lie algebra g, finite-dimensional over a field k of characteristic 0.  ...  -3 ring Q, which is a two-sided maximal quotient ring, with minimal faithful modules Qe and fQ such that F is a subring of Q containing Qe and fQ.  ...

### Algebraic Dependence of Arithmetic Functions

J. Popken
1962 Indagationes Mathematicae (Proceedings)
with the multiplication in the ring R.  ...  Such a ring (e.g. an integral domain of characteristic zero) shall be called torsionfree.Theorem 2. Let R be torsion-free. Let /1, /2, ... , fr be algebraically dependent over R'.  ...

### Variations on a theme of Minkowski and Serre

A. Silverberg, Yu.G. Zarhin
1996 Journal of Pure and Applied Algebra
If CL is a root of unity in an integral domain 0 of characteristic zero, (a -l)k E no, and no prime divisor of n is a unit in 0, then a = 1 if n is a positive integer outside a finite set determined by  ...  Notation All rings in this paper are rings with identity. However, we do not assume 0 # 1; that is, we do not exclude the zero ring.  ...  Then cxRckP) = 1; in particular, CC= 1 ifn\$N(k). Rings in characteristic zero Remark 4.1.  ...

### Page 1261 of Mathematical Reviews Vol. 46, Issue 5 [page]

1973 Mathematical Reviews
If P is a generator in the category of right A-modules such that the ring B is semiprime, then P is of finite type and projective. J. Inh (Raleigh, N.C.)  ...  One also finds there Brauer’s useful result that, if R is a field, the class of M in G,*(A) is determined by the characteristic polynomials on M of the elements of A.  ...

### Page 618 of Mathematical Reviews Vol. 46, Issue 3 [page]

1973 Mathematical Reviews
Authors’ summary: “Let F be a finite extension of the p-adic rational field Q,, R, a ring of integers in F and A,(R,G) an algebra of R,-representations of a finite group G over the rational field Q.  ...  No. 40 (1971), 81- 116] by considering families 6={G(q)}, of type (W, R), where (W, R) is a finite Coxeter system, q runs through a fixed infinite set €P (= “characteristic powers”) of prime powers and  ...

### Trace and defect of a block idempotent

Gerhard O Michler
1990 Journal of Algebra
DEDICATED TO WALTER FEIT ON THE OCCASION OF HIS 60TH BIRTHDAY In the study of group rings SG of groups G over fields S of characteristic zero the trace tr(a) of an idempotent a = CgeG s,g, SUE S, is defined  ...  In fact by Theorem 2.1(b) tr(i) = ~"-~x/y, where x and y are integers which are not divisible by p. Since for every idempotent QE SG the trace tr(a) = IGI -' dim.  ...  visit in 1988.  ...

### A note on reduced Jordan algebras

Kevin McCrimmon
1968 Proceedings of the American Mathematical Society
Since p\ei= UPie}e2 = UV{ei)ese2=Ue1Ue2Ue1e2=pip2piei we have p2(pi-p2)=0; similarly P2(P2-Pi)=0, so either pi-p2 = 0 or Pi=p2 = 0, and in either case Pi=p2=p.  ...  In this section we make no assumptions about the simplicity or finite-dimensionality of A. Recall that an idempotent e is reduced if Ai(e) = UeA=\$e.  ...

### A Note on Reduced Jordan Algebras

Kevin McCrimmon
1968 Proceedings of the American Mathematical Society
Since p\ei= UPie}e2 = UV{ei)ese2=Ue1Ue2Ue1e2=pip2piei we have p2(pi-p2)=0; similarly P2(P2-Pi)=0, so either pi-p2 = 0 or Pi=p2 = 0, and in either case Pi=p2=p.  ...  In this section we make no assumptions about the simplicity or finite-dimensionality of A. Recall that an idempotent e is reduced if Ai(e) = UeA=\$e.  ...

### A Design and an Implementation of an Inverse Kinematics Computation in Robotics Using Real Quantifier Elimination based on Comprehensive Gröbner Systems [article]

Shuto Otaki, Akira Terui, Masahiko Mikawa
2021 arXiv   pre-print
Furthermore, pre-computation of CGS and substituting parameters in the CGS with the given values avoids the repetitive computation of Gr\"obner basis.  ...  Then, the quotient ring K[ X]/I is regarded as finite dimensional vector space over K  ; let {v 1 , . . . , v d } be its basis.  ...  In Step P2, we choose F ⊂ F satisfying that S lj contains real points and the system of polynomial equations in eq. ( 12 ) has finite number of roots for c ∈ S lj .  ...

### Torsion-Free Abelian Group Rings III

Ryuki MATSUDA
1977 Bulletin of the Faculty of Science Ibaraki University Series A Mathematics
Assume G be finitely generated and _??_ a prime ideal of qKG[X]=qe in KG[X]. Then, _??_•¿ kG[X]=P. PROOF. By Prop. 3, qe: _??_ce=(q: _??_c)e. Hence, By Prop. 1, Hence, _??  ...  Let F'G'•¸qe for F', G'•¸S and G'•¸qe. We may suppose F' , G' E k[t]G[X]. By  (6.15), qR[t] is primary in R[t].  ...

### The Brauer-Wall group of a commutative ring

Charles Small
1971 Transactions of the American Mathematical Society
The Brauer-Wall group is therefore a fundamental tool towards a general theory of quadratic forms (over an arbitrary commutative ring). This theme is developed in [3, Chapter V].  ...  Brauer group of a commutative ring, separable algebra, Azumaya algebra, graded algebra, Galois extension of commutative rings, quadratic extension of commutative rings. 0) This paper contains part of the  ...  When A has characteristic 2, A{a} is a QE of A: for any a g A:, and in fact all QE's are of this form ; the crossed product described above is then m When 2 g Uik), \k(a)\ is a QE for any a e ¿/(A), and  ...

### The Brauer-Wall Group of a Commutative Ring

Charles Small
1971 Transactions of the American Mathematical Society
The Brauer-Wall group is therefore a fundamental tool towards a general theory of quadratic forms (over an arbitrary commutative ring). This theme is developed in [3, Chapter V].  ...  Brauer group of a commutative ring, separable algebra, Azumaya algebra, graded algebra, Galois extension of commutative rings, quadratic extension of commutative rings. 0) This paper contains part of the  ...  When A has characteristic 2, A{a} is a QE of A: for any a g A:, and in fact all QE's are of this form ; the crossed product described above is then m When 2 g Uik), \k(a)\ is a QE for any a e ¿/(A), and  ...

### F-jumping and F-Jacobian ideals for hypersurfaces [article]

Luis Núñez-Betancourt, Felipe Pérez
2013 arXiv   pre-print
We introduce two families of ideals, F-jumping ideals and F-Jacobian ideals, in order to study the singularities of hypersurfaces in positive characteristic.  ...  In addition, we use F-Jacobian ideals to study intrinsic properties of the singularities of hypersurfaces. In particular, we give conditions for F-regularity.  ...  Let R be a reduced ring essentially of finite type over an excellent local ring of prime characteristic.  ...

### On rings whose Morita class is represented by matrix rings

Piercarlo Merisi, Peter Vámos
1998 Journal of Pure and Applied Algebra
In the description of rings of this type, Picard progenerators and (in the commutative case) the Picard group play a significant role.  ...  In a number of special cases these two conditions will also become sufficient.  ...  We are grateful to Ken Goodearl for drawing our attention to the class of ultramatricial von Neumann regular rings as a useful source of examples of MM rings and to Roger Wiegand for helpfkl conversations  ...

### Quaternion constituents of group algebras

Mark Benard
1971 Proceedings of the American Mathematical Society
In this paper it is shown that each quaternion division algebra central over the rationals appears as a division ring constituent of some rational group algebra.  ...  Let x be an irreducible character of a finite group and let A be a field of characteristic 0.  ...  To compute mQp(d) we use a theorem of Kronstein  Thus we have mQq(x) = 2 ii q = p or oo and mQq(x) = 1 if qE {2, p, oo }. In particular wq(x) = 2 and the constituent Z?  ...
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