In algebra, an analytically irreducible ring is a local ring whose completion has no zero divisors. Geometrically this corresponds to a variety with only...

local ring that is analytically reducible and therefore not analytically normal. Nagata, Masayoshi (1958), "An example of a normal local ring which is...

example of an analytically ramified reduced local ring. Krull showed that every 1-dimensional normal Noetherian local ring is analytically unramified; more...

x − 1 ) ( x − 2 ) {\displaystyle y^{2}-x(x-1)(x-2)} is an irreducible polynomial. The ring Z [ x ] / ( x 2 − n ) ≅ Z [ n ] {\displaystyle \mathbb {Z}...

that 3 would not be irreducible. For D > 0, ω is a positive irrational real number, and the corresponding quadratic integer ring is a set of algebraic...

tensor product of irreducible representations is not irreducible; the process of decomposing a tensor product as a direct sum of irreducible representations...

(SGA 1 §XII. Proposition 2.1.) Complex analytic variety (or just variety) is sometimes required to be irreducible and (or) reduced Aroca, José Manuel; Hironaka...

rings contains several other counterexamples he found, such as a commutative Noetherian ring that is not catenary, and a commutative Noetherian ring of...

irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology. Under this definition, non-irreducible...

ideal of the polynomial ring. Some authors do not make a clear distinction between algebraic sets and varieties and use irreducible variety to make the distinction...

\ldots ,a_{n}\in K} . As another example, an algebraic subset W in Kn is irreducible (in the Zariski topology) if and only if I ( W ) {\displaystyle I(W)}...

where Zi are the irreducible components of the scheme-theoretic intersection of X and H with multiplicity (length of the local ring) mi. A complex projective...

local ring is called analytically unramified if its completion has no nonzero nilpotent elements. 3.  A local ring is called analytically irreducible if...

topological space, meaning the supremum of the lengths of all chains of irreducible closed subsets. This follows immediately from the Galois connection between...

is a normal point of a variety then it is analytically normal; in other words the completion of the local ring at x is a normal integral domain (Mumford...

curve is irreducible, then one has an irreducible plane algebraic curve. Otherwise, the algebraic curve is the union of one or several irreducible curves...

element is irreducible if it is neither a unit nor the product of two other non-unit elements. In the ring of integers, the prime and irreducible elements...

multiplicity-free (or spherical variety) if every irreducible representation of G appears at most one time in the coordinate ring; i.e., dim ⁡ k [ X ] ( λ ) ≤ dim ⁡...

\mathbb {R} } , f {\displaystyle f} will generally no longer be irreducible, but its irreducible (real) factors are either of degree one or two. Since there...

polynomial ( 1 + x ) {\displaystyle (1+x)} is irreducible in the polynomial ring, and hence the code is an irreducible code. The idempotent of this code is the...

ring Z[√-5]. In this ring, the numbers 3, 2 + √-5 and 2 - √-5 are irreducible. This means that the number 9 has two factorizations into irreducible elements...

polynomial ring F[u] for a finite field F, one can ask how often a finite set of polynomials fi(x) in F[u][x] simultaneously takes irreducible values in...

nonzero element of ( R / m ) {\displaystyle (R/{\mathfrak {m}})} and irreducible polynomials that are monic (that is, their leading coefficients are 1)...

L2(S1). The irreducible representations of all compact connected Lie groups have been classified. In particular, the character of each irreducible representation...

varieties, and let r = dim X − dim Y. Then For every irreducible closed subset W of Y and every irreducible component Z of f − 1 ( W ) {\displaystyle f^{-1}(W)}...

algebraic function is a function that can be defined as the root of an irreducible polynomial equation. Algebraic functions are often algebraic expressions...

Let X be a normal irreducible projective scheme over a discrete valuation ring. If the generic fiber is ruled, then each irreducible component of the special...

the polynomial x 8 + u 3 {\displaystyle x^{8}+u^{3}\,} over the ring F2[u] is irreducible and has no fixed prime polynomial divisor (after all, its values...

be a principal ideal domain, K be its field of fractions, and π be an irreducible element of R. Since every principal ideal domain is a unique factorization...

( ρ , s ) {\displaystyle L(\rho ,s)} of a non-trivial irreducible representation ρ is analytic in the whole complex plane. This is known for one-dimensional...