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

* Every polynomial ring R ..., x < sub > n </ sub > is a commutative R-algebra.

Every Boolean algebra ( A, ∧, ∨) gives rise to a ring ( A, +, ·) by defining a + b := ( a ∧ ¬ b ) ∨ ( b ∧ ¬ a ) = ( a ∨ b ) ∧ ¬( a ∧ b ) ( this operation is called symmetric difference in the case of sets and XOR in the case of logic ) and a · b := a ∧ b. The zero element of this ring coincides with the 0 of the Boolean algebra ; the multiplicative identity element of the ring is the 1 of the Boolean algebra.

In Norse mythology, Draupnir ( Old Norse " the dripper ") is a gold ring possessed by the god Odin with the ability to multiply itself: Every ninth night eight new rings ' drip ' from Draupnir, each one of the same size and weight as the original.

Every module over a division ring has a basis ; linear maps between finite-dimensional modules over a division ring can be described by matrices, and the Gaussian elimination algorithm remains applicable.

Every division ring is therefore a division algebra over its center.

Every objective has a different size ring, so for every objective another condenser setting has to be chosen.

Every match must be assigned a rule keeper known as a referee, who is the final arbitrator ( In multi-man lucha libre matches, two referees are used, one inside the ring and one outside ).

* In any ring R, a maximal ideal is an ideal M that is maximal in the set of all proper ideals of R, i. e. M is contained in exactly 2 ideals of R, namely M itself and the entire ring R. Every maximal ideal is in fact prime.

* Krull's theorem ( 1929 ): Every ring with a multiplicative identity has a maximal ideal.

Every Boolean ring R satisfies x ⊕ x

Every prime ideal P in a Boolean ring R is maximal: the quotient ring R / P is an integral domain and also a Boolean ring, so it is isomorphic to the field F < sub > 2 </ sub >, which shows the maximality of P. Since maximal ideals are always prime, prime ideals and maximal ideals coincide in Boolean rings.

Every finitely generated ideal of a Boolean ring is principal ( indeed, ( x, y )=( x + y + xy )).

* Every finitely-generated commutative algebra over a commutative Noetherian ring is Noetherian.

* Every localization of a commutative Noetherian ring is Noetherian.

Every topological ring is a topological group ( with respect to addition ) and hence a uniform space in a natural manner.

Every prostaglandin contains 20 carbon atoms, including a 5-carbon ring.

Every polynomial in can be factorized into polynomials that are irreducible over F. This factorization is unique up to permutation of the factors and the multiplication of the factors by nonzero constants from F ( because the ring of polynomials over a field is a unique factorization domain whose units are the nonzero constant polynomials ).

Every such subset has a smallest element, so to specify our choice function we can simply say that it maps each set to the least element of that set.

** Well-ordering theorem: Every set can be well-ordered.

Every information exchange between living organisms — i. e. transmission of signals that involve a living sender and receiver can be considered a form of communication ; and even primitive creatures such as corals are competent to communicate.

Every context-sensitive grammar which does not generate the empty string can be transformed into an equivalent one in Kuroda normal form.

* Every regular language is context-free because it can be described by a context-free grammar.

Every grammar in Chomsky normal form is context-free, and conversely, every context-free grammar can be transformed into an equivalent one which is in Chomsky normal form.

Every real number has a ( possibly infinite ) decimal representation ; i. e., it can be written as

Every entire function can be represented as a power series that converges uniformly on compact sets.

Group actions / representations: Every group G can be considered as a category with a single object whose morphisms are the elements of G. A functor from G to Set is then nothing but a group action of G on a particular set, i. e. a G-set.

Every positive integer n > 1 can be represented in exactly one way as a product of prime powers:

Every sequence can, thus, be read in three reading frames, each of which will produce a different amino acid sequence ( in the given example, Gly-Lys-Pro, Gly-Asn, or Glu-Thr, respectively ).

Every hyperbola is congruent to the origin-centered East-West opening hyperbola sharing its same eccentricity ε ( its shape, or degree of " spread "), and is also congruent to the origin-centered North-South opening hyperbola with identical eccentricity ε — that is, it can be rotated so that it opens in the desired direction and can be translated ( rigidly moved in the plane ) so that it is centered at the origin.

Every holomorphic function can be separated into its real and imaginary parts, and each of these is a solution of Laplace's equation on R < sup > 2 </ sup >.

Every species can be given a unique ( and, one hopes, stable ) name, as compared with common names that are often neither unique nor consistent from place to place and language to language.

Every vector v in determines a linear map from R to taking 1 to v, which can be thought of as a Lie algebra homomorphism.

Every morpheme can be classified as either free or bound.

Every use of modus tollens can be converted to a use of modus ponens and one use of transposition to the premise which is a material implication.

Every document window is an object with which the user can work.

Every adult, healthy, sane Muslim who has the financial and physical capacity to travel to Mecca and can make arrangements for the care of his / her dependants during the trip, must perform the Hajj once in a lifetime.

Every ordered field can be embedded into the surreal numbers.

* Every finite topological space gives rise to a preorder on its points, in which x ≤ y if and only if x belongs to every neighborhood of y, and every finite preorder can be formed as the specialization preorder of a topological space in this way.

* Every preorder can be given a topology, the Alexandrov topology ; and indeed, every preorder on a set is in one-to-one correspondence with an Alexandrov topology on that set.

Every binary relation R on a set S can be extended to a preorder on S by taking the transitive closure and reflexive closure, R < sup >+=</ sup >.