Ideal quotient information

In abstract algebra, if I and J are ideals of a commutative ring R, their ideal quotient (I : J) is the set ( I : J ) = { r ∈ R ∣ r J ⊆ I } {\displaystyle...

universal algebra. Starting with a ring R and a two-sided ideal I in R, a new ring, the quotient ring R / I, is constructed, whose elements are the cosets...

fractional ideals. In this case, the fractional ideal J {\displaystyle J} is uniquely determined and equal to the generalized ideal quotient ( R : K I...

there are no other ideals contained between I and R. Maximal ideals are important because the quotients of rings by maximal ideals are simple rings, and...

field of quotients, or quotient field of R {\displaystyle R} . All four are in common usage, but are not to be confused with the quotient of a ring by...

commutator subalgebra is an ideal in g {\displaystyle {\mathfrak {g}}} , in fact the smallest ideal such that the quotient Lie algebra is abelian. It is...

primitive ideals are always two-sided ideals. Primitive ideals are prime. The quotient of a ring by a left primitive ideal is a left primitive ring. For commutative...

coefficients. The previous example can be further exploited by taking quotients from prime ideals. For example, the ring C [ x , y ] / ( y 2 − x ( x − 1 ) ( x...

called the quotient semigroup or factor semigroup, and denoted S / ~. The mapping x ↦ [x]~ is a semigroup homomorphism, called the quotient map, canonical...

object, and the quotient of additive categories defined above coincides in this case with the notion of a quotient ring modulo a two-sided ideal. The localization...

for defining the quotient is not of the same nature as the ambient space (that is, a quotient ring is the quotient of a ring by an ideal, not a subring...

Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both...

fractional ideals. The quotient of the group of non-zero fractional ideals by this subgroup is the ideal class group. Two fractional ideals I and J represent...

coequalizer of two ring homomorphisms f and g from R to S is the quotient of S by the ideal generated by all elements of the form f(r) − g(r) for r ∈ R. Given...

remainder theorem may be stated as: if m and n are coprime integers, the quotient ring Z / m n Z {\displaystyle \mathbb {Z} /mn\mathbb {Z} } is the product...

algebra over any field. The total quotient ring of the zero ring is itself. For any ring A and ideal I of A, the quotient A/I is the zero ring if and only...

integers, Z, then the norm of a nonzero ideal I of a number ring R is simply the size of the finite quotient ring R/I. Let A be a Dedekind domain with...

In abstract algebra, the total quotient ring or total ring of fractions is a construction that generalizes the notion of the field of fractions of an integral...

\mathbb {C} \langle x,y\rangle /(xy-yx-1)} , where the ideal corresponds to the commutator The quotient ring C ⟨ x 1 , … , x n ⟩ / ( x i x j − q i j x j x...

If I is an ideal of S then f−1(I) is an ideal of R. If R and S are commutative and P is a prime ideal of S then f−1(P) is a prime ideal of R. If R and...

positive integers. It is usually denoted GR(pn, r). It can be defined as a quotient ring GR ⁡ ( p n , r ) ≅ Z [ x ] / ( p n , f ( x ) ) {\displaystyle \operatorname...

\left(R\setminus \left\{0\right\}\right)^{-1}R} is a field, called the quotient field of R {\displaystyle R} . Many of the following notions also exist...

is always a Dedekind domain, and so has unique factorization of ideals into prime ideals. The units of a ring of integers OK is a finitely generated abelian...

the ideal class group (or class group) of an algebraic number field K is the quotient group JK /PK where JK is the group of fractional ideals of the...

proper realm to study divisibility. Principal ideal domains are integral domains in which every ideal can be generated by a single element, another property...