Every irreducible element of R is also irreducible element of R

This seems to be a statement rather than a question. However, I can explain the concept behind it.

In ring theory, a branch of abstract algebra, an element in a commutative ring is said to be irreducible if it is non-zero, not a unit, and cannot be expressed as a product of two non-units.

The statement "Every irreducible element of R is also irreducible element of R" is a tautology, meaning it is always true by definition. If an element is irreducible in a ring R, it remains irreducible in the same ring R.

Here are the steps to understand this:

1. Understand the definition of an irreducible element: An element 'a' in a ring 'R' is called irreducible if it is non-zero, not a unit, and cannot be expressed as a product of two non-units.

2. Apply this definition to the statement: If an element is irreducible in R, it means it satisfies all the conditions of the definition in the context of R.

3. Therefore, it is obvious that the same element will still satisfy all these conditions in the same ring R. Hence, it remains irreducible in R.

So, the statement is always true.