some of the so called smallness conditions in algebra as well as in category theory, are important and interesting for their own and also tightly related to injectivity, are essential boundedness, cogenerating set, and residual smallness.in this overview paper, we first try to refresh these smallness condition by giving the detailed proofs of the results mainly by bernhard banaschewski and walt...

Actively growing turfgrasses usually contain more than 75 percent water. Their “dry” matter holds 16 nutrient elements considered essential for survival, flowering and/or seed production. Turfgrasses receive three of these, carbon (C), hydrogen (H) and oxygen (O), from carbon dioxide in the air and from water. The remaining essential elements are minerals, most often supplied by the soil. Healt...

Some of the so called smallness conditions in algebra as well as in category theory, are important and interesting for their own and also tightly related to injectivity, are essential boundedness, cogenerating set, and residual smallness. In this overview paper, we first try to refresh these smallness condition by giving the detailed proofs of the results mainly by Bernhard Banaschewski and W...

Essential extensions and p-extensions have been studied for commutative rings with identity in various articles, such as [2], [3], and [13]. The present article applies these concepts to certain subrings of C(X). Moreover, the article introduces a new ring extension, called a pg-extension, and determines its relation to both essential extension and p-extension. It turns out that the pg-extensio...