** Every unital ring other than the trivial ring contains a maximal ideal.

Considered as a ring, however, it has only the trivial automorphism.

* The spectrum of any commutative ring with the Zariski topology ( that is, the set of all prime ideals ) is compact, but never Hausdorff ( except in trivial cases ).

To exclude the trivial ring, the additive identity and the multiplicative identity are required to be distinct.

Wedderburn's little theorem states that the Brauer group of a finite field is trivial, so that every finite division ring is a finite field.

In abstract algebra, an integral domain is a commutative ring that has no zero divisors, and which is not the trivial ring

There are zero objects also for the category of abelian groups, category of pseudo-rings Rng ( trivial ring ), category of modules over a ring, and category of vector spaces over a field ; see zero object ( algebra ) for details.

The trivial ring consisting only of a single element 0 = 1 is a terminal object.

The empty scheme ( equal to the prime spectrum of the trivial ring ) is an initial object.

The biproduct condition in the case n = 0 simplifies drastically ; B is a nullary biproduct if and only if the identity morphism of B is the zero morphism from B to itself, or equivalently if the hom-set Hom ( B, B ) is the trivial ring.

Graded modules may be considered over non-graded rings by giving the trivial gradation to the ring.

In the usual case where the ring R is not graded ( in particular if R is a field ), it is given the trivial grading ( every element of " R " is of grade 0 ).

For example, the class group of a Dedekind domain is trivial if and only if the ring is a unique factorization domain.

The size of the ideal class group can be considered as a measure for the deviation of a ring from being a principal domain ; a ring is a principal domain if and only if it has a trivial ideal class group.

This is a group homomorphism ; its kernel is the group of units of R, and its cokernel is the ideal class group of R. The failure of these groups to be trivial is a measure of the failure of the map to be an isomorphism: that is the failure of ideals to act like ring elements, that is to say, like numbers.

Modern texts include the axiom 0 ≠ 1 to avoid having to consider the trivial ring or a " field with one element ", where the multiplicative identity coincides with the additive identity.

Since η is a ring morphism, then one must have either that A is the trivial ring, or that η is injective.

This ring is called the trivial ring.

Alternatively, linear independence can be directly defined as follows: a set of vectors is linearly independent if and only if the only representations of the zero vector as linear combinations of its elements are trivial solutions, i. e., whenever a < sub > 1 </ sub >, a < sub > 2 </ sub >, ..., a < sub > n </ sub > are scalars such that

Philosopher and sociologist Theodor Adorno defined schadenfreude as “ largely unanticipated delight in the suffering of another which is cognized as trivial and / or appropriate ”.

These specific deficits, termed aphasias, may be defined as impairments of language production or comprehension that cannot be attributed to trivial causes such as deafness or oral paralysis.

They are sometimes defined as compact Kähler manifolds whose canonical bundle is trivial, though many other similar but inequivalent definitions are sometimes used.

The standard method to construct the reciprocity homomorphism is to first construct the local reciprocity isomorphism from the multiplicative group of the completion of a global field to the Galois group of its maximal abelian extension ( this is done inside local class field theory ) and then prove that the product of all such local reciprocity maps when defined on the idele group of the global field is trivial on the image of the multiplicative group of the global field.

In mathematics, a trivial ring is a ring defined on a singleton set,

If K is a field then the quiver algebra or path algebra KΓ is defined as a vector space having all the paths ( of length ≥ 0 ) in the quiver as basis ( including, for each vertex i of the quiver Γ, a trivial path of length 0 ; these paths are not assumed to be equal for different i ), and multiplication given by concatenation of paths.

Here ω < sup > α </ sup >< sub > β </ sub > defines a k × k matrix of one-forms on U. In fact, given any such matrix the above expression defines a connection on E restricted to U. This is because ω < sup > α </ sup >< sub > β </ sub > determines a one-form ω with values in End ( E ) and this expression defines ∇ to be the connection d + ω, where d is the trivial connection on E over U defined by differentiating the components of a section using the local frame.

In mathematics, the Frattini subgroup Φ ( G ) of a group G is the intersection of all maximal subgroups of G. For the case that G has no maximal subgroups, for example the trivial group e or the Prüfer group, it is defined by Φ ( G ) = G. It is analogous to the Jacobson radical in the theory of rings, and intuitively can be thought of as the subgroup of " small elements " ( see the " non-generator " characterization below ).

However, if we now assume that grass is green on the left-hand side refers to a proposition, then the theory seems trivial since snow is white is defined as true if and only if snow is white.

The trivial module must be one dimensional, and so an algebra homomorphism may be defined such that for all v in.

Finally, we let T < sup >∗</ sup > y = z, completing the construction of T < sup >∗</ sup >.< ref > Verifying that T < sup >∗</ sup > is linear trivial .</ ref > Note that T < sup >∗</ sup > exists if and only if T is densely defined.

The Tate – Shafarevich group of an abelian variety A defined over a number field K consists of the elements of the Weil – Châtelet group that become trivial in all of the completions of K.

The trivial MV-algebra has the only element 0 and the operations defined in the only possible way, and

The glass may seem trivial but Communist official hooliganism feeds on such incidents unless they are redressed.

Linguists have not always been more enlightened than `` practical people '' and sometimes have insisted on incredibly trivial points while neglecting things of much greater significance.

Depending on the underlying logic, the problem of deciding the validity of a formula varies from trivial to impossible.

AMD's older CPUs could simply be set to run at whatever clock speed the user chose on the motherboard, making it trivial to relabel a CPU and sell it as a faster grade than it was originally intended.

Clement criticizes Greek paganism in the Protrepticus on the basis that its deities are both false and poor moral examples, and he attacks the mystery religions for their obscurantism and trivial rituals.

In measurements of purely mechanical systems ( involving units of length, mass, force, energy, pressure, and so on ), the differences between CGS and SI are straightforward and rather trivial ; the unit-conversion factors are all powers of 10 arising from the relations and.

Likewise, the problem of computing a quantity on a manifold which is invariant under differentiable mappings is inherently global, since any local invariant will be trivial in the sense that it is already exhibited in the topology of R < sup > n </ sup >.

According to Suetonius, some were convicted for corruption or treason, others on trivial charges, which Domitian justified through his suspicion:

The equality equivalence relation is the finest equivalence relation on any set, while the trivial relation that makes all pairs of elements related is the coarsest.

Meanwhile on Usenet, Mark Horton had started a series of " Periodic Posts " ( PP ) which attempted to answer trivial questions with appropriate answers.

The design of the HMAC specification was motivated by the existence of attacks on more trivial mechanisms for combining a key with a hash function.

The trivial name is assigned based on historical precedent or popular usage rather than a standardized naming convention.

The condition on the fundamental group turns out to be necessary ( and sufficient ) for finite time extinction, and in particular includes the case of trivial fundamental group.

He emphasized the need to " enforce the leading role of the party more effectively " and acknowledged that, despite Klement Gottwald's urgings for better relations with society, the Party had too often made heavy-handed rulings on trivial issues.

Apart from the trivial zeros, the Riemann zeta function doesn't have any zero on the right of σ = 1 and on the left of σ = 0 ( neither the zeros can lie too close to those lines ).

The property of separability does not in and of itself give any limitations on the cardinality of a topological space: any set endowed with the trivial topology is separable, as well as second countable, quasi-compact, and connected.

Topelius initially thought of writing a trivial entertainment, but having heard extracts from the opera project at a concert in 1851, he realized that Pacius was writing a grand opera on the theme of salvation, following the early Romantic style of Carl Maria von Weber's Der Freischütz ( 1821 ) and Oberon ( 1826 ).

The trivial case is checking to see that a file opened, written to, or closed, did not fail on these activities – which is a sanity check often ignored by programmers.

He said the two allegations of misconduct were trivial administrative issues and had nothing to do with the discovery of bubble nuclear fusion or the underlying science, and that " all allegations of fraud and fabrication have been dismissed as invalid and without merit — thereby supporting the underlying science and experimental data as being on solid ground ".

A criticism of this approach is that it is trivial, based on the trivial observation that mathematics is a human activity.

Thus, exports minus imports — which is the GDP formula — declined from 1 billion to 600 million, while GDP was $ 58. 9 billion, resulting in a trivial effect on GDP of about two-thirds of one percent.

This more permissive approach is not however the standard when considering formal power series, as it would lead to convergence considerations that are as subtle as they are in analysis, while the philosophy of formal power series is on the contrary to make convergence questions as trivial as they can possibly be.

Software running on these systems can use an alternative separator ( such as ";", or "-"), and it is often trivial to patch free and open source software to use a different separator.