MATH 250B: ALGEBRA SEMISIMPLICITY 1. Remarks on non-commutative rings
ثبت نشده
چکیده
We begin with some reminders and remarks on non-commutative rings. First, recall that if R is a not necessarily commutative ring, an element x ∈ R is invertible if it has both a left (multiplicative) inverse and a right (multiplicative) inverse; it then follows that the two are necessarily equal. Note however, that unlike the example of n× n matrices over a field, in a general noncommutative ring having a left inverse does not imply the existence of a right inverse, and vice versa. When R is not commutative, we will use the term “module” over R to mean “left module” over R. Furthermore, the standard definitions of matrices and matrix multiplication all work perfectly over non-commutative rings, and in particular given a non-commutative ring R, we have the ring Matn(R) of n×n matrices with coefficients in R. We say that R is a division ring if every nonzero elements is invertible; that is, a division ring is a field which is not necessarily commutative. Then, much of the theory of vector spaces over a field extends to modules over a division ring: in particular, they are all free, and any two bases of a given module have the same cardinality. We will still refer to modules over division rings as “vector spaces”, and the cardinality of a basis as the “dimension.”
منابع مشابه
On the commuting graph of non-commutative rings of order $p^nq$
Let $R$ be a non-commutative ring with unity. The commuting graph of $R$ denoted by $Gamma(R)$, is a graph with vertex set $RZ(R)$ and two vertices $a$ and $b$ are adjacent iff $ab=ba$. In this paper, we consider the commuting graph of non-commutative rings of order pq and $p^2q$ with Z(R) = 0 and non-commutative rings with unity of order $p^3q$. It is proved that $C_R(a)$ is a commutative ring...
متن کاملOn the commuting graph of some non-commutative rings with unity
Let $R$ be a non-commutative ring with unity. The commuting graph of $R$ denoted by $Gamma(R)$, is a graph with a vertex set $Rsetminus Z(R)$ and two vertices $a$ and $b$ are adjacent if and only if $ab=ba$. In this paper, we investigate non-commutative rings with unity of order $p^n$ where $p$ is prime and $n in lbrace 4,5 rbrace$. It is shown that, $Gamma(R)$ is the disjoint ...
متن کاملMath 101b: Algebra Ii Part C: Semisimplicity
We have one week to talk about semisimple rings and semisimple modules (Chapter XVII). A semisimple R-module is a finite direct sum of simple modules M = S1 ⊕ · · · ⊕ Sn and a semisimple ring is a ring R for which all f.g. modules are semisimple. The main reasons that I am choosing this particular topic in noncommutative algebra is for the study of representations of finite groups which we will...
متن کاملA Note on Perturbation Theory for Semi-groups of Operators
1. I. S. Cohen, Commutative rings with restricted minimum conditions, Duke Math. J. 17 (1950), 27-41. 2. S. MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771-782. 3. N. Nakano, Idealtheorie in einem speziellen unendlichen algebraischen Zahlkorper, J. Sei. Hiroshima Univ. 16(1953), 425-439. 4. O. Zariski and P. Samuel, Commutative a...
متن کامل