SYARAT PERLU DAN CUKUP RING LOKAL KOMUTATIF AGAR RING MATRIKSNYA BERSIH KUAT (?-REGULAR KUAT); THE NECESSARY AND SUFFICIENT CONDITIONS OF COMMUTATIVE LOCAL RINGS SUCH THAT THE MATRIX RINGS ARE STRONGLY CLEAN (STRONGLY ?-REGULAR)
ANAS YOGA NUGROHO, Budi Surodjo
2014 | Disertasi | PROGRAM STUDI S2 MATEMATIKARing R is called ring strongly ?-regular if every element of R is left ?- regular and right ?-regular. Furthermore, every element of strongly ?-regular can be expressed as the sum of units and idempotent element which commute with the multiplication operation. As a result, every element of strongly ?-regular is an elemen of strongly clean. Commutative local ring is one example of a strongly clean ring, but not necessarily strongly ?-regular ring. Based on the research that has been done, the necessary and sufficient condition of a commutative local ring R said to be a strongly ?-regular ring is if J(R) is a Nil ideal. If commutative local ring R with J(R) Nil ideal then the matrix ring Mn(R) is a strongly ?-regular ring. On the other hand, if the commutative local ring R is an n-SRC ring then matrix ring Mn(R) is strongly clean ring.
Kata Kunci : COMMUTATIVE LOCAL RINGS, MATRIX RINGS
