For example, in some groups, the group operation is commutative, and this can be asserted with the introduction of an additional axiom, but without this axiom we can do quite well developing ( the more general ) group theory, and we can even take its negation as an axiom for the study of non-commutative groups.

For the case of a non-commutative base ring R and a right module M < sub > R </ sub > and a left module < sub > R </ sub > N, we can define a bilinear map, where T is an abelian group, such that for any n in N, is a group homomorphism, and for any m in M, is a group homomorphism too, and which also satisfies

For square matrices with entries in a non-commutative ring, for instance the quaternions, there is no unique definition for the determinant, and no definition that has all the usual properties of determinants over commutative rings.

For the element A ( t ) from the algebra ( set g with the non-commutative product *), where t is the " time parameter ", the ordered exponential

For this reason, some talk about non-commutative topology, though the term has also other meanings.

For the duality between σ-finite measure spaces and commutative von Neumann algebras, noncommutative von Neumann algebras are called non-commutative measure spaces.

For a blood-chilling ring of terror to the very sound of his name was the tool he needed for the job he'd promised to do.

For bridged alkenes, the Bredt's rule states that a double bond cannot be placed at the bridgehead of a bridged ring system, unless the rings are large enough.

For example, the proof that the column rank of a matrix over a field equals its row rank yields for matrices over division rings only that the left column rank equals its right row rank: it does not make sense to speak about the rank of a matrix over a division ring.

For example, a benzene ring or double bond in conjugation with the carbonyl will bring the wavenumber down about 30 cm < sup >− 1 </ sup >.

* K, the ring of polynomials over a field K. For each nonzero polynomial P, define f ( P ) to be the degree of P.

* K < nowiki ></ nowiki > X < nowiki ></ nowiki >, the ring of formal power series over the field K. For each nonzero power series P, define f ( P ) as the degree of the smallest power of X occurring in P. In particular, for two nonzero power series P and Q, f ( P )≤ f ( Q ) iff P divides Q.

For other examples of quotient objects, see quotient ring, quotient space ( linear algebra ), quotient space ( topology ), and quotient set.

For example, a ring possesses both addition and multiplication, and a homomorphism from the ring to the ring is a function such that

For example, a Hall sensor integrated into a ferrite ring ( as shown ) can reduce the detection of stray fields by a factor of 100 or better ( as the external magnetic fields cancel across the ring, giving no residual magnetic flux ).

For example, most network cards implement transmit and receive ring buffers, interrupt rate limiting, and hardware flow control.

For the end, two finalists squared off in a boxing ring for a two-minute round to declare the champion.

For example, the aldohexose glucose may form a hemiacetal linkage between the hydroxyl on carbon 1 and the oxygen on carbon 4, yielding a molecule with a 5-membered ring, called glucofuranose.

For example, in the ring, we have

For instance, electrons and positive ions trapped in the dipole-like field near the Earth tend to circulate around the magnetic axis of the dipole ( the line connecting the magnetic poles ) in a ring around the Earth, without gaining or losing energy ( this is known as Guiding center motion ).

For example, hemoglobin and myoglobin contain an iron center coordinated to the nitrogen atoms of a porphyrin ring ; magnesium is the center of a chlorin ring in chlorophyll.

For instance, the ring ( in fact field ) of complex numbers, which can be constructed from the polynomial ring R over the real numbers by factoring out the ideal of multiples of the polynomial.

For example, prime ideals in the ring of integers of quadratic number fields can be used in proving quadratic reciprocity, a statement that concerns the solvability of quadratic equations

For safety, the ropes are padded at the turnbuckles and cushioned mats surround the floor outside the ring.

For instance, the determinant of the matrix

For example, if the affine transformation acts on the plane and if the determinant of A is 1 or − 1 then the transformation is an equi-areal mapping.

For this to be an invariant, the epsilon symbol must be a tensor, and so must contain a factor of √ g, where g is the determinant of the metric tensor.

For a general n × n invertible matrix A, i. e., one with nonzero determinant, A < sup >− 1 </ sup > can thus be written as an ( n − 1 )- th order polynomial expression in A: As indicated, the Cayley – Hamilton theorem amounts to the identity

For Sp ( 2 ), the group of 2 x 2 matrices with determinant 1, the three symplectic ( 0, 1 )- matrices are:

For a 2 × 2 matrix A, the characteristic polynomial can be found from its determinant and trace, tr ( A ), to be

For most industries, the intensity of competitive rivalry is the major determinant of the competitiveness of the industry.

For Easterlin, the size of the cohort is a critical determinant of how easy it is to get a good job.

For functions of more than one variable, the theorem states that if the total derivative of a continuously differentiable function F defined from an open set U of R < sup > n </ sup > into R < sup > n </ sup > is invertible at a point p ( i. e., the Jacobian determinant of F at p is non-zero ), then F is an invertible function near p. That is, an inverse function to F exists in some neighborhood of F ( p ).

For n functions of several variables, a generalized Wronskian is the determinant of an n by n matrix with entries D < sub > i </ sub >( f < sub > j </ sub >) ( with 0 ≤ i < n ), where each D < sub > i </ sub > is some constant coefficient linear partial differential operator of order i. If the functions are linearly dependent then all generalized Wronskians vanish.

For the determinant and trace, the following properties hold

For an N-electron system, the Slater determinant is defined as

For instance, the determinant of a tridiagonal matrix A of order n can be computed by the recursive formula for a continuant

For A a commutative ring, one can define a determinant det: GL ( A ) → A < sup >*</ sup > to the group of units of A, which vanishes on E ( A ) and thus descends to a map det: K < sub > 1 </ sub >( A ) → A < sup >*</ sup >.

For quiescent probiotic bacteria, water activity is a crucial determinant of survival in food products during storage.

For a regular surface, this determinant is non-vanishing ; equivalently, the Jacobian matrix has rank 2.

For equally ( balanced ) correlation of ground and excited states ( better excitation energies ) one can use more than one reference determinant from which all singly, doubly, ... excited determinants are included ( multireference configuration interaction ).

For nearly degenerate determinants which build the ground state one should use the multi-configurational self-consistent field ( MCSCF ) method because the Hartree – Fock determinant is qualitatively wrong and so are the CI wave functions and energies.

For instance, the Linder hypothesis states that demand plays a more important role than comparative advantage as a determinant of trade -- with the hypothesis that countries which share similar demands will be more likely to trade.

* For any cocycle C, the determinant det C ( x, t ) is a one-dimensional cocycle.

For instance, this allows for the computation of the determinant of the Laplace and Dirac operators on a Riemannian manifold, using the Minakshisundaram – Pleijel zeta function.