The same definition holds in any unital ring or algebra where a is any invertible element.

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.

These antibiotics have a common element in their molecular structure: a four-atom ring known as a beta-lactam.

Let A be a unital commutative Banach algebra over C. Since A is then a commutative ring with unit, every non-invertible element of A belongs to some maximal ideal of A.

A ring homomorphism of commutative rings determines a morphism of Kähler differentials which sends an element dr to d ( f ( r )), the exterior differential of f ( r ).

In some guns, this element is replaced with a wide ring and is called a muzzle band.

If A is a fixed element of a ring ℜ, the first additional relation can also be interpreted as a Leibniz rule for the map given by B ↦.

Specifically, it is a non-trivial ring in which every non-zero element a has a multiplicative inverse, i. e., an element x with.

Some authors also require the domain of the Euclidean function be the entire ring R ; this can always be accommodated by adding 1 to the values at all nonzero elements, and defining the function to be 0 at the zero element of R, but the result is somewhat awkward in the case of K. The definition is sometimes generalized by allowing the Euclidean function to take its values in any well-ordered set ; this weakening does not affect the most important implications of the Euclidean property.

In abstract algebra, a field is a commutative ring which contains a multiplicative inverse for every nonzero element, equivalently a ring whose nonzero elements form an abelian group under multiplication.

This ring is a field because it contains a multiplicative inverse for each element N other than zero ( an integer that, multiplied by the element modulo p yields 1 ), and it has a finite number of elements ( p ), making it a finite field.

If R is a commutative ring, and a and b are in R, then an element d of R is called a common divisor of a and b if it divides both a and b ( that is, if there are elements x and y in R such that d · x = a and d · y = b ).

In a ring all of whose ideals are principal ( a principal ideal domain or PID ), this ideal will be identical with the set of multiples of some ring element d ; then this d is a greatest common divisor of a and b. But the ideal ( a, b ) can be useful even when there is no greatest common divisor of a and b. ( Indeed, Ernst Kummer used this ideal as a replacement for a gcd in his treatment of Fermat's Last Theorem, although he envisioned it as the set of multiples of some hypothetical, or ideal, ring element d, whence the ring-theoretic term.

The counterweight is then thrown from hand to hand and used as an additional element in the trick.

Also, whereas most of these were about who did it ( Inspector Morse or Taggart ) and why it was done ( Cracker ), this new series would be about how it was done with an element of magic thrown in, such as murders committed in locked rooms, a person being in two places at once or impossible thefts ; finding a culprit would still be part of the detective's job but the emphasis would be on how the crime was committed.

:.... With the risk of not being understood by some traditionalists which may be among you, I am in favour of the forced migration of the entire Jew element from Bessarabia and Bukovina, which must be thrown over the border.

Steinberg obviously has concluded that it is the lyric element which must dominate in this score, and he manages at times to create the effect of the whole orchestra bursting into song.

The element of public danger which enters so largely into architectural certification, however, would demand a prolonged study of structure.

Astatine is an extremely radioactive element ; all its isotopes have half-lives of less than 12 hours, decaying into bismuth, polonium, radon, or other astatine isotopes.

* Coast Guard-the maritime element of the defence force, and is divided into four units:

Steven Harper states " Wesley does not place the substitionary element primarily within a legal framework ... Rather doctrine seeks to bring into proper relationship the ' justice ' between God's love for persons and God's hatred of sin ... it is not the satisfaction of a legal demand for justice so much as it is an act of mediated reconciliation.

This was not a cynical use of religion to manipulate his subjects into obedience, but an intrinsic element in Alfred's worldview.

A major meta-plot element was the sealing off of the aforementioned Renraku Arcology in Seattle when the advanced computer control system awakened into a self-aware AI named Deus.

When analyzing the efficiency of algorithms that use stacks, one may also specify that all operations take the same time no matter how many items have been pushed into the stack, and that the stack uses a constant amount of storage for each element.

This extension is needed for, among other things, modeling the concepts of " is an element of " or " is a subset of " in set theory, without running into logical inconsistencies such as Russell's paradox.

Just as kets and bras can be transformed into each other ( making into ) the element from the dual space corresponding with is where A < sup >†</ sup > denotes the Hermitian conjugate ( or adjoint ) of the operator A.

Any Banach algebra ( whether it has an identity element or not ) can be embedded isometrically into a unital Banach algebra so as to form a closed ideal of.

What we have done here is arranged the integers and the odd integers into a one-to-one correspondence ( or bijection ), which is a function that maps between two sets such that each element of each set corresponds to a single element in the other set.

It might seem natural to divide the sets into different classes: put all the sets containing one element together ; all the sets containing two elements together ; ...; finally, put together all infinite sets and consider them as having the same size.

Cumry is an evoluted form of the Old Welsh, with an assimilation of to first, the second element brogi changed into bro “ country ” in Modern Welsh.

A dynamical system is a manifold M called the phase ( or state ) space endowed with a family of smooth evolution functions Φ < sup > t </ sup > that for any element of t ∈ T, the time, map a point of the phase space back into the phase space.

The third element incorporated into many definitions is that of intentionality – the death must be intended, rather than being accidental, and the intent of the action must be a " merciful death ".

While people expected elements with higher atomic number to form from neutron bombardment of lighter elements, nobody expected neutrons to have enough energy to actually split a heavier atom into two light element fragments, and it was thought still more unlikely that slow neutrons could accomplish such a task.

These areas do not stand alone, however, as often a particular item or element may fit into more than one of these areas.

It can be noted that a generation of 16 fermions can be put into the form of an Octonion with each element of the octonion being an 8-vector.

This setting of the novel, a castle or religious building, often one fallen into disrepair, was an essential element of the Gothic novel.

It was barely energetically favorable for helium to fuse into the next element with a lower energy per nucleon, carbon.

By sorting formulas according to the number of atoms of each element present in the formula according to these rules, with differences in earlier elements or numbers being treated as more significant than differences in any later element or number — like sorting text strings into lexicographical order — it is possible to collate chemical formulas into what is known as Hill system order.

The moment the sea closed over Nick, some atavistic sense warned him that he would survive in this alien element only if he did not panic.

Silicon is an essential element in biology, although only tiny traces of it appear to be required by animals, however various sea sponges as well as microorganisms like diatoms need silicon in order to have structure.

The Athenian people, facing the gravest moment of peril in their history, committed themselves once and for all to the alien element of the sea, and put their faith in a man whose ambitions many had long profoundly dreaded.

The " Mero -" or " Mer -" element in the name suggests a sea or ocean ( see Old English " mere ," Latin " mare ," or even the Modern English word " mermaid ", etc .).

The representatives of the family can be found in nearly every habitat, including dense or open forests, in scrublands, on wet fields and valleys, at the sea coast and in marine areas, and in swamps and as an element of mangrove woods.

The first element in his name is cognate with the stem of the name of the Irish sea god Manannán mac Lir, and likely originated from the same Celtic deity as Manannán.

Medieval Irish scholars thought the name contained the element muire " sea ", owing to their reputation as sea pirates.

In a 19th-century work, Paul Henri Mallet and Walter Scott write that the " fen " element of Fensalir " may also be made to sig the watery deep, or the sea.

With darkness the sea was inseparably connected by the Celts, and, as soon as the dark twin was born and named, he plunged headlong into his native element ... Beautiful legends grew up around his death.

The first element is brim which means " breaker wave " or " heavy sea ".

The first element is probably derived from the word mosk / musk which means " sea spray " and the last element is nes which means " headland ".

In the circum-Baltic and more westwards coastal Scandinavian areas, there is clear evidence of a maritime economy, where the sea has to be seen as a uniting element.

Arguments in favor of this method generally relate to the ability to target specific areas that may not necessarily be easily accessible by land or sea, a greater chance of surprising the enemy and overwhelming defensive structures, and, in many cases, the need for a reduced number of forces due to the element of surprise.

Iodine, an element essential for human health, is present only in negligible amounts in sea salt, albeit the concentration varies according to its provenance.

Although Operation Dynamo at Dunkirk had evacuated much of the fighting element of the British Expeditionary Force, some combat units from 1st Armoured Division and Beauman Division and more than 150, 000 support and line-of-communication troops had been cut-off to the south by the German “ dash to the sea ”.

In the navy the battleship, long seen as the dominant element of sea power, was displaced by the greater range and striking power of the aircraft carrier.

The town's name means either " Skeggi's headland " or " beard-shaped headland ", depending on whether the first element represents the personal name Skeggi ( meaning ' bearded one '), one of the Vikings who established the original settlement to the east of the current town which was washed away by the sea in the early sixteenth century ; or the Old East Norse word skegg " beard ".

The axi element is also validated by the older Greek name for the Black Sea, Ἄξεινος πόντος – Axeinos pontos, later altered to the euphemism Εὔξεινος πόντος Euxeinos pontos meaning " Hospitable sea ".

Joyce in his Irish Names of Places states that it is probable that in the first element of " Olderfleet " is the ancient Irish name of the Larne water, Ollorbha, with the second element being a corruption of the word " fiord " ( a long narrow arm of the sea, running up between high banks or cliffs, as on the coast of Norway )

Military Sealift Command's Prepositioning Program is an element in the US's triad of power projection into the 21st century-sea shield, sea strike and sea basing.