2.1. The Basic Consequence about Polynomial Theory over R
. Obviously, we know that
For the sake of convenience, set . Notice that and . Let with then
It is indicated that this special non-chain ring can be considered as a 2-dimensional algebra over with being its basis. The significance of this basis is that v, w are idempotent and orthogonal.
Define two canonical projective maps The above facts illustrate that and are -algebra homomorphism. Similarly, for all , it can be seen that are -algebra homomorphism.
As the application to polynomial theory, extend this thought from R
to polynomial ring
, it is clear that
Consequently, define the two maps still are -algebra homomorphism. For simplicity, if one mathematical object appears to subscript v or w, it uses the projective maps or by default.
It follows from the above illustration that . For , write . This guarantees that the element of can be viewed as the element of .
Next, let us consider the divisibility between any two elements in . Then, the following proposition holds.
For , let with . Then, in if and only if in .
For , let . So . Then, . Due to the expression by the basis , it follows that and , which means that .
Let , then and in . □
Proposition 1 implies that and for any with .
Although is not a field, it inherits the nice properties related to , particularly with regard to factorization. Therefore, let us consider the greatest common divisor between any two elements in . Similarly, the following proposition is obtained.
Proposition 2. For , let with . Then, in ,
where the symbol on the right hand side of the equation denotes the greatest common divisor in .
Let . Clearly, . From Proposition 1, holds. For the same reason, also holds. These indicate that is a common divisor of and in .
For every with , in terms of Proposition 1, write , where , and . As a result of the conventional polynomial theory over finite fields, and are obtained. Applying Proposition 1 again obtains .
Hence, holds. □
According to the above proof, and also holds.
The above analysis indicated that for
, one has
, which derives that
2.3. Further Results about Polynomial Theory over R
Definition 1. Let C be an R-submodule of , C is called a double cyclic code of length over R if implies that
Denote by the set of all double cyclic codes of length over R.
For , let be the coordinate projection of C on the first m coordinates, and be the coordinate projection of C on the second n coordinates. These mean that are R-linear map and , (For the convenience of writing, one mathematical object has the subscript m or n, which also means that it used by the coordinates projected to m or n). Hence, if and only if and .
For simplicity, , are called canonical projective maps, and , are called coordinate projective maps.
A code is separable if C is the direct product of and .
Just like the situation of cyclic codes over finite fields, there exists a bijection between
Let this bijective map expressed by . Set
Then the rings
with this action, which is induced by the action of
from the multiplication of
, become the
-module. Simultaneously, define two maps
Then, are still -module homomorphism.
This subsection reveals the fact that if and only if is a -submodule to . Hence, the issue of -submodule of needs to be of concern in this paper. Based on the bijection of , double cyclic codes over R as the -submodule of will be studied.