Sentence examples for be a local ring from inspiring English sources

Let A be a local ring with non-zero maximal ideal ({mathfrak {m}}).

Theorem 2 Let R be a local ring, and the set of the non-units is denoted by I.

Let A be a local ring with non-zero maximal ideal ({mathfrak {m}}) such that ({rm Ann}({mathfrak {m}})ne 0).

Let R be a local ring and the set of the non-units is denoted by I. Now we recall a theorem and corollary which are constructed in [2] for Moufang-Klingenberg planes.

(Theorem XIII.7 of [[8]]) Let (A M) be a local ring and a(x) be a regular polynomial in A[x] such that Π(a(x)) is irreducible in K [ x ] Open image in new window.

(Theorem XIII.2 of [[8]]) Let (A M) be a local ring and a ( x ) = ∑ i = 0 n a i x i ∈ A [ x ] Open image in new window.

Q is a local ring (and it is called the dual local ring on Q).

Moreover, if A is a local ring, then ({rm diam}({mathbb {G}}(A))le 2).

By Theorem 8, if A is a local ring, then the cases (ii) or (iii) occur.

If A is a local ring, then ({mathbb {G}}(A)) is a star and so ({mathbb {G}}(A)) is bipartite.

Moreover, if A is a local ring and ({mathbb {G}}(A)) contains a cycle, then ({rm gr}({mathbb {G}}(A))=3).