Gray arsenic ( α-As, space group Rm No. 166 ) adopts a double-layered structure consisting of many interlocked ruffled six-membered rings.

Metallic antimony adopts a layered structure ( space group Rm No. 166 ) in which layers consist of fused ruffled six-membered rings.

In the context of abstract algebra, for example, a mathematical object is an algebraic structure such as a group, ring, or vector space.

In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.

Upon heating, α-berkelium transforms into another phase with an fcc lattice ( but slightly different from β-berkelium ), space group Fmm and the lattice constant of 500 pm ; this fcc structure is equivalent to the closest packing with the sequence ABC.

Watterson opposed the structure publishers imposed on Sunday newspaper cartoons: the standard cartoon starts with a large, wide rectangle featuring the cartoon's logo or a throwaway panel tangential to the main area so that newspapers pressed for space can remove the top third of the cartoon if they wish ; the rest of the strip is presented in a series of rectangles of different widths.

Binary trees can also be stored in breadth-first order as an implicit data structure in arrays, and if the tree is a complete binary tree, this method wastes no space.

A succinct data structure is one which takes the absolute minimum possible space, as established by information theoretical lower bounds.

( The exact structure of this Hilbert space depends on the situation.

If in the above we relax Banach space to normed space the analogous structure is called a normed algebra.

Since a maximal ideal in A is closed, is a Banach algebra that is a field, and it follows from the Gelfand-Mazur theorem that there is a bijection between the set of all maximal ideals of A and the set Δ ( A ) of all nonzero homomorphisms from A to C. The set Δ ( A ) is called the " structure space " or " character space " of A, and its members " characters.

In some cases, the details of electronic structure are less important than the long-time phase space behavior of molecules.

It was this notion of compactness that became the dominant one, because it was not only a stronger property, but it could be formulated in a more general setting with a minimum of additional technical machinery, as it relied only on the structure of the open sets in a space.

Raw CMBR data coming down from the space vehicle ( i. e., WMAP ) contain foreground effects that completely obscure the fine-scale structure of the Cosmic Microwave background.

Instead, with the topology of compact convergence, C ( a, b ) can be given the structure of a Fréchet space: a locally convex topological vector space whose topology can be induced by a complete translation-invariant metric.

Since Cauchy sequences can also be defined in general topological groups, an alternative to relying on a metric structure for defining completeness and constructing the completion of a space is to use a group structure.

Since the corona has been photographated at high resolution in the X-rays by the satellite Skylab in 1973, and then later by Yohkoh and the other following space instruments, it has been seen that the structure of the corona is very various and complex: different zones have been immediately classified on the coronal disc

In physical terms, dimension refers to the constituent structure of all space ( cf.

This observation motivates the theoretical concept of an abstract data type, a data structure that is defined indirectly by the operations that may be performed on it, and the mathematical properties of those operations ( including their space and time cost ).

These conditions played an important role in the development of the structure theory of commutative rings in the works of David Hilbert, Emmy Noether, and Emil Artin.

Prime numbers give rise to two more general concepts that apply to elements of any commutative ring R, an algebraic structure where addition, subtraction and multiplication are defined: prime elements and irreducible elements.

In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec ( R ), is the set of all proper prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space.

The notion of a Noetherian ring is of fundamental importance in both commutative and noncommutative ring theory, due to the role it plays in simplifying the ideal structure of a ring.

The inner structure of a commutative ring is determined by considering its ideals, i. e. nonempty subsets that are closed under multiplication with arbitrary ring elements and addition: for all r in R, i and j in I, both ri and i + j are required to be in I.

* The Laurent polynomial ring RX < sup >− 1 </ sup > is isomorphic to the group ring of the group Z of integers over R. More generally, the Laurent polynomial ring in n variables is isomorphic to the group ring of the free abelian group of rank n. It follows that the Laurent polynomial ring can be endowed with a structure of a commutative, cocommutative Hopf algebra.

In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field ( or, more generally over a commutative ring ) with an extra layer of structure, known as a gradation ( or grading ).

* Algebra over a ring ( also R-algebra ): a free module over a commutative ring R, which also carries a ring structure that is compatible with the module structure.

If K is only a commutative ring and not a field, then the same process works if A is a free module over K. If it isn't, then the multiplication is still completely determined by its action on a set that spans A ; however, the structure constants can't be specified arbitrarily in this case, and knowing only the structure constants does not specify the algebra up to isomorphism.

As a torus, J carries a commutative group structure, and the image of C generates J as a group.

Hence, by the structure theorem for finitely generated abelian groups, it is isomorphic to a product of a free abelian group Z < sup > r </ sup > and a finite commutative group for some non-negative integer r called the rank of the abelian variety.

* The Zariski-Samuel theorem determines the structure of a commutative principal ideal rings.

Reversing all the arrows in the commutative diagrams that define a Lie supergroup then shows that functions over the supergroup have the structure of a Z < sub > 2 </ sub >- graded Hopf algebra.

A noncommutative algebra is an associative algebra in which the multiplication is not commutative, that is, for which xy does not always equal yx ; or more generally an algebraic structure in which one of the principal binary operations is not commutative ; one also allows additional structures, e. g. topology or norm to be possibly carried by the noncommutative algebra of functions.

Just as the R-modules are central in commutative algebra when studying the commutative ring R, so are the O < sub > X </ sub >- modules central in the study of the scheme X with structure sheaf O < sub > X </ sub >.

This formula is multilinear over N in each variable ; and so defines a ring structure on the tensor product, making into a commutative N-algebra, called the tensor product of fields.

* If R is commutative, the matrix ring has a structure of a *- algebra over R, where the involution * on M < sub > n </ sub >( R ) is the matrix transposition.

They concern a number of interrelated ( sometimes surprisingly so ) conjectures relating various homological properties of a commutative ring to its internal ring structure, particularly its Krull dimension and depth.

For an arbitrary group scheme G, the ring of global sections also has a commutative Hopf algebra structure, and by taking its spectrum, one obtains the maximal affine quotient group.

In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an ( unital associative ) algebra and a ( counital coassociative ) coalgebra, with these structures ' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property.

This construction passes from the non-associative structure L to a ( more familiar, and possibly easier to handle ) unital associative algebra which captures the important properties of L.

The noncommutative ring theory, besides the rich structure theory, includes the study of rings such as Banach algebras and operator algebras in functional analysis and the representation ring and cohomology rings in geometry.

* Hilbert space: an inner product space over the real or complex numbers whose inner product gives rise to a Banach space structure.

Even though the topological structure of Fréchet spaces is more complicated than that of Banach spaces due to the lack of a norm, many important results in functional analysis, like the Hahn – Banach theorem, the open mapping theorem, and the Banach – Steinhaus theorem, still hold.

Fubini's theorem also shows that this operator is continuous with respect to the Banach space structure on L < sup > 1 </ sup >, and that the following inequality holds:

The reason is that K-theory behaves much better on topological algebras such as Banach algebras or C *- algebras than on algebras without additional structure.