minor doc tweaks

docs/src/CommutativeAlgebra/ca.md

@@ -13,7 +13,7 @@ Pages = ["ca.md"]
 # Introduction
 
 The commutative algebra part of OSCAR provides functionality for handling
-- ideals and modules over multivariated polynomial rings,
+- ideals and modules over multivariate polynomial rings,
 - quotient rings of such rings, with particular emphasis on affine $K$-algebras.
 
 It is under development with regard to providing both the functionality and the documentation. 

docs/src/CommutativeAlgebra/ca_affine_algebras.md

@@ -251,7 +251,7 @@ $A = \bigoplus_{d\geq 0} A_d$, where each graded piece $A_d$ is a finite dimensi
 $K$-vector space. In this situation, the *Hilbert function* of $A$ is
 the function
 
-$H(A, \underline{\phantom{d}}): \N \rightarrow \N, d \mapsto \dim_K(d).$
+$H(A, \underline{\phantom{d}}): \N \to \N, d \mapsto \dim_K(d).$
 
 The *Hilbert series* of $A$ is the generating function
 

docs/src/CommutativeAlgebra/ca_modules.md

@@ -15,7 +15,7 @@ Pages = ["ca_modules.md"]
 Modules over a multivariate polynomial ring are implemented as `subquotients`.
 That is, they are  submodules of quotients of free modules. Explicitly, the subquotient
 $M$ with generators $A$ and relations $B$ over the ring $R$ is defined to be the module
-$M = (\text{im } A + \text{im }  B)/\text{im }  B$, where $A: R^m\rightarrow R^p$ and $B: R^m\rightarrow R^p$
+$M = (\text{im } A + \text{im }  B)/\text{im }  B$, where $A: R^m\to R^p$ and $B: R^m\to R^p$
 are two matrices representing maps of free $R$-modules with the same codomain.
 
 !!! note

docs/src/Groups/autgroup.md

@@ -22,7 +22,7 @@ automorphism_group(G)
 ```
 The evaluation of the automorphism `f` in the element `x` is analogous to the homomorphism evaluation: it can be obtained by typing either `f(x)` or `x^f`.
 
-It is possible to turn an automorphism `f` into a homomorphism by typing `hom(f)`. The viceversa is also possible: if `g` is a bijective homomorphism from the group `G` to itself and `A` is the automorphism group of `G`, then the instruction `A(g)` returns `g` as automorphism of `G`. This is the standard way to build explicitly an automorphism (another way, available for inner automorphisms, is shown in Section [Inner_automorphisms](@ref inner_automorphisms)).
+It is possible to turn an automorphism `f` into a homomorphism by typing `hom(f)`. The converse is also possible: if `g` is a bijective homomorphism from the group `G` to itself and `A` is the automorphism group of `G`, then the instruction `A(g)` returns `g` as automorphism of `G`. This is the standard way to build explicitly an automorphism (another way, available for inner automorphisms, is shown in Section [Inner_automorphisms](@ref inner_automorphisms)).
 
 ```@docs
 automorphism_group(G::GAPGroup)

docs/src/InvariantTheory/it.md

@@ -29,7 +29,7 @@ is its ring of invariants.
 
 !!! note
     Except where mentioned otherwise, we will be in the favourable situation where $G$ is a linear reductive group which acts rationally on $V$. This has several important consequences:
-    - There exists a Reynolds operator $\mathcal R: K[V] \rightarrow K[V]$. That is, $\mathcal R$ is a $K$-linear graded map which projects $K[V]$ onto $K[V]^G$, and which is a $K[V]^G$-module homomorphism;
+    - There exists a Reynolds operator $\mathcal R: K[V] \to K[V]$. That is, $\mathcal R$ is a $K$-linear graded map which projects $K[V]$ onto $K[V]^G$, and which is a $K[V]^G$-module homomorphism;
     - by Hilbert's finiteness theorem, $K[V]^G$ is finitely generated as a $K$-algebra;
     - by a result of Hochster and Roberts, $K[V]^G$ is Cohen-Macaulay. Equivalently, $K[V]^G$ is a free module (of finite rank) over any of its Noether normalizations.
     If $k[V]^G$ is finitely generated as a $K$-algebra, we call any irredundant system of homogeneous generators a fundamental system of invariants of $k[V]^G$. By Nakayama's lemma, the number of elements in such a system is uniquely determined as the embedding dimension of $K[V]^G$. Similarly, the degrees of these elements are uniquely determined.

docs/src/Rings/integer.md

@@ -546,7 +546,7 @@ rising_factorial(ZZ(-30), 3)
 * `primorial(n::Base.Int) -> Base.Int`
 * `primorial(n::Oscar.Integer) -> Oscar.Integer`
 
-Returns the promorial ``P(n)``, i.e. the product of all primes less than or
+Returns the primorial ``P(n)``, i.e. the product of all primes less than or
 equal to ``n``. An exception is raised if ``n < 0``. We define
 ``P(0) = P(1) = 1``.