सुबरी इसके अंग्रेजी अर्थ का उदाहरण

If S is a subring of a ring R, then M \otimes_R N is the quotient group of M \otimes_S N by the subgroup generated by xr \otimes_S y - x \otimes_S ry, \, r \in R, x \in M, y \in N, where x \otimes_S y is the image of (x, y) under \otimes: M \times N \to M \otimes_{S} N.

It is enough to show that there is a subring S of A that is generated by m-1 elements, such that A is finite over S.

Let G be a finite group and let Char(G) denote the subring of the ring of complex-valued class functions of G consisting of integer combinations of irreducible characters.

The name could stem from that of the Canturigi, a population of Insubria of the 6th century BC.

In , it was proven that a ring which is the direct sum of two nilpotent subrings is itself nilpotent.

The sum of a nilpotent subring and a nil subring is always nil.

A locally nilpotent ring is one in which every finitely generated subring is nilpotent: locally nilpotent rings form a radical class, giving rise to the Levitzki radical.

Since it is well known that each integral domain is a subring of a field of fractions (via an embedding) in such a way that every element is of the form rs−1 with s nonzero, it is natural to ask if the same construction can take a noncommutative domain and associate a division ring (a noncommutative field) with the same property.

For every right Ore domain R, there is a unique (up to natural R-isomorphism) division ring D containing R as a subring such that every element of D is of the form rs−1 for r in R and s nonzero in R.

Any domain satisfying one of the Ore conditions can be considered a subring of a division ring, however this does not automatically mean R is a left order in D, since it is possible D has an element which is not of the form s−1r.

Another natural question is: "When is a subring of a division ring right Ore?" One characterization is that a subring R of a division ring D is a right Ore domain if and only if D is a flat left R-module .

A subdomain of a division ring which is not right or left Ore: If F is any field, and G \langle x,y \rangle\, is the free monoid on two symbols x and y, then the monoid ring F[G]\, does not satisfy any Ore condition, but it is a free ideal ring and thus indeed a subring of a division ring, by .

Let F be the field of fractions of R, and put , the subring of polynomials in F[X] with constant term in R.