## What do you mean by polynomial rings?

In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field.

**Is polynomial ring a free module?**

The polynomial ring R[X] is a free R-module with basis 1, X, X2,… . We will adopt the standard convention that the zero module is free with the empty set as basis.

**What is Laurent polynomial ring?**

The ring of Laurent polynomials R[X, X−1] is an extension of the polynomial ring R[X] obtained by “inverting X”. More rigorously, it is the localization of the polynomial ring in the multiplicative set consisting of the non-negative powers of X.

### Is polynomial ring a local ring?

The ring of formal power series k[[X1…Xn]] over a field k or over any local ring is local. On the other hand, the polynomial ring k[X1…Xn] with n≥1 is not local.

**What are polynomial rings and polynomial codes?**

Polynomial Rings are analagous to the ring of integers. A polynomial p(x) is divisible by a polynomial q(x) if there exists a polynomial r(x) such that p(x) = q(x)r(x). The polynomials q(x) and r(x) are also called factors of p(x).

**What are the units in a polynomial ring?**

Let (F,+,∘) be a field whose zero is 0F and whose unity is 1F. Let F[X] be the ring of polynomial forms in an indeterminate X over F. Then the units of F[X] are all the elements of F[X] whose degree is 0.

## How can I see which module is not free?

In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commutative case), then there exist non-free modules.

**What is the rank of a module?**

The rank of a free module M over an arbitrary ring R( cf. Free module) is defined as the number of its free generators. For rings that can be imbedded into skew-fields this definition coincides with that in 1). In general, the rank of a free module is not uniquely defined.

**Is a principal ideal domain?**

A principal ideal domain is an integral domain in which every proper ideal can be generated by a single element. The term “principal ideal domain” is often abbreviated P.I.D. Examples of P.I.D.s include the integers, the Gaussian integers, and the set of polynomials in one variable with real coefficients.

### Are polynomial rings vector spaces?

Ring of Polynomial Forms over Field is Vector Space.

**Is Z projective module?**

An R-module P is a projective module if there exists an R- module Q such that P ⊕ Q is a free R-module. 43.3 Examples. 1) If R is a ring with identity then every free R-module is projective. 2) Z/2Z and Z/3Z are non-free projective Z/6Z-modules.