diff --git a/src/main/scala/org/appliedtopology/tda4j/Chain.scala b/src/main/scala/org/appliedtopology/tda4j/Chain.scala
index d89f2547..fb7799ce 100644
--- a/src/main/scala/org/appliedtopology/tda4j/Chain.scala
+++ b/src/main/scala/org/appliedtopology/tda4j/Chain.scala
@@ -21,77 +21,10 @@ import scala.annotation.targetName
   * @param chainMap
   *   Internal storage of the sorted map of the elements
   */
-class Chain[CellT <: Cell[CellT]: Ordering, CoefficientT]
+class Chain[CellT <: Cell[CellT]: Ordering, CoefficientT : Fractional]
 /** chainMap is an immutable variable and constructor that uses Scala's SortedMap to make a key-value pairing of an CellT as the key and a
   * CoefficientT type as the value. Here, we'll use the Using keyword to check for any relevant types for CoefficientT.
-  */ (val chainMap: SortedMap[CellT, CoefficientT])(using
-  fr: Fractional[CoefficientT]
-) {
-
-  /** Negate takes in a Chain instance and outputs a new Chain instance which
-    * takes in a chainMap (remember chainMap is using SortedMap 'under the
-    * hood', thus already has a key/value pair). transform() is used on this
-    * function's instance of chainMap to transform the key/value pairings in
-    * chainMap to the negative versions of the values in the value pairs.
-    * @return
-    *   new Chain of negated values in key/value pairs
-    */
-  def negate: Chain[CellT, CoefficientT] = new Chain(
-    chainMap.transform((k, v) => fr.negate(v))
-  )
-
-  /** Unary uses negate() for unary negation
-    *
-    * @return
-    *   negation
-    */
-  def unary_- : Chain[CellT, CoefficientT] = negate
-
-  /** scalarMultiply returns a new chain containing chainMap and its key/value
-    * pairing. In the chainMap, transform() is used on the value of the
-    * key/value pairing. On the value, it is transformed by multiplying each key
-    * by the CoefficientT, using the times() method of the Fractional trait,
-    * extending the Numeric library.
-    * @param c:
-    *   method instance of CoefficientT
-    * @return
-    *   new Chain with values in key/value pairing multiplied by c
-    */
-
-  def scalarMultiply(c: CoefficientT): Chain[CellT, CoefficientT] =
-    new Chain(chainMap.transform((k, v) => fr.times(v, c)))
-
-  def *: : CoefficientT => Chain[CellT, CoefficientT] = scalarMultiply
-
-  /** add() adds the method instance of the keys of a chainMap to the classes
-    * chainMap keys. It then adds the result of this to a map which maps that
-    * key to the values of the class and the instance method's chainMap.
-    * @param that:
-    *   Chain object composed of a Cell/Coefficient pair
-    * @return
-    *   Chain object composed of a Cell/Coefficient pair
-    */
-  def add(that: Chain[CellT, CoefficientT]): Chain[CellT, CoefficientT] =
-    Chain(
-      (chainMap.keySet | that.chainMap.keySet)
-        .map(k =>
-          (
-            k,
-            fr.plus(
-              chainMap.getOrElse(k, fr.zero),
-              that.chainMap.getOrElse(k, fr.zero)
-            )
-          )
-        )
-        .toSeq*
-    )
-
-  def + : Chain[CellT, CoefficientT] => Chain[CellT, CoefficientT] = add
-
-  def subtract(that: Chain[CellT, CoefficientT]): Chain[CellT, CoefficientT] =
-    this + (-that)
-
-  def - : Chain[CellT, CoefficientT] => Chain[CellT, CoefficientT] = subtract
+  */ (val chainMap: SortedMap[CellT, CoefficientT]) {
 
   override def toString: String =
     chainMap.map((k, v) => s"${v.toString} *: ${k.toString}").mkString(" + ")
@@ -142,12 +75,37 @@ object Chain {
   // Original apply innards
   // new Chain[CellT, CoefficientT](SortedMap.from(items))
 
-  def apply[CellT <: Cell[CellT]: Ordering, CoefficientT](cell: CellT)(using
+  def apply[CellT <: Cell[CellT] : Ordering, CoefficientT](cell: CellT)(using
     fr: Fractional[CoefficientT]
   ): Chain[CellT, CoefficientT] =
     new Chain[CellT, CoefficientT](SortedMap.from(List(cell -> fr.one)))
 }
 
+
+class ChainOps[CellT <: Cell[CellT] : Ordering, CoefficientT : Fractional] extends RingModule[Chain[CellT, CoefficientT], CoefficientT] {
+  import Numeric.Implicits._
+
+  val zero: org.appliedtopology.tda4j.Chain[CellT, CoefficientT] = Chain()
+
+  def plus(x: Chain[CellT, CoefficientT], y: Chain[CellT, CoefficientT]): Chain[CellT, CoefficientT] =
+    Chain(
+      (for k <- (x.chainMap.keySet | y.chainMap.keySet)
+        yield {
+          val fr = summon[Fractional[CoefficientT]]
+          val xv : CoefficientT = x.chainMap.getOrElse(k, fr.zero)
+          val yv : CoefficientT = y.chainMap.getOrElse(k, fr.zero)
+          k -> fr.plus(xv,yv)
+        }).toSeq : _*
+    )
+
+  def scale(x: CoefficientT, y: Chain[CellT, CoefficientT]): Chain[CellT, CoefficientT] =
+    new Chain(y.chainMap.transform((k,v) => x*v))
+
+  override def negate(x: Chain[CellT, CoefficientT]): Chain[CellT, CoefficientT] =
+    new Chain(x.chainMap.transform((k, v) => -v))
+}
+
+
 /** Lightweight trait to define what it means to be a topological "Cell".
   *
   * Using F-bounded types to ensure reflective typing. See
diff --git a/src/main/scala/org/appliedtopology/tda4j/RingModule.scala b/src/main/scala/org/appliedtopology/tda4j/RingModule.scala
new file mode 100644
index 00000000..590a0cf3
--- /dev/null
+++ b/src/main/scala/org/appliedtopology/tda4j/RingModule.scala
@@ -0,0 +1,36 @@
+package org.appliedtopology.tda4j
+
+import scala.annotation.targetName
+
+/** Specifies what it means for the type `T` to be a module (or vector space)
+ * over the [Numeric] (ie ring-like) type `R`.
+ *
+ * @tparam T Type of the module elements.
+ * @tparam R Type of the ring coefficients
+ */
+trait RingModule[T, R] {
+  val zero: T
+  def plus(x: T, y: T): T
+  def minus(x: T, y: T): T = plus(x, negate(y))
+  def negate(x: T): T = minus(zero, x)
+  def scale(x: R, y: T): T
+
+  extension (t: T) {
+    @targetName("add")
+    def +(rhs: T): T = plus(t, rhs)
+    @targetName("subtract")
+    def -(rhs: T): T = minus(t, rhs)
+    @targetName("scalarMultiplyRight")
+    def <*(rhs: R): T = scale(rhs, t)
+    def unary_- = negate(t)
+  }
+
+  extension (r: R) {
+    @targetName("scalarMultiplyLeft")
+    def *>(t: T): T = scale(r, t)
+  }
+}
+
+object RingModule:
+  def apply[T,R](using rm: RingModule[T,R]) = rm
+
diff --git a/src/main/scala/org/appliedtopology/tda4j/Simplex.scala b/src/main/scala/org/appliedtopology/tda4j/Simplex.scala
index 5228c877..93732f83 100644
--- a/src/main/scala/org/appliedtopology/tda4j/Simplex.scala
+++ b/src/main/scala/org/appliedtopology/tda4j/Simplex.scala
@@ -17,6 +17,30 @@ import scala.math.Ordering.IntOrdering
 import scala.math.Ordering.Double.IeeeOrdering
 import math.Ordering.Implicits.sortedSetOrdering
 
+trait SimplexContext[VertexT : Ordering] {
+  type Simplex = AbstractSimplex[VertexT]
+
+  given Ordering[Simplex] = sortedSetOrdering[AbstractSimplex, VertexT]
+
+  object Simplex {
+    def apply(vertices: VertexT*): Simplex = AbstractSimplex[VertexT](vertices : _*)
+    def empty: Simplex = AbstractSimplex.empty
+    def from(iterableOnce: IterableOnce[VertexT]): Simplex =
+      AbstractSimplex.from(iterableOnce)
+    def newBuilder: mutable.Builder[VertexT, Simplex] = AbstractSimplex.newBuilder
+  }
+
+  object s {
+    def apply(vertices: VertexT*): Simplex = Simplex(vertices: _*)
+  }
+
+  extension (s: Simplex) {
+    def className = "Simplex"
+  }
+}
+
+
+
 /** Type alias creating `Simplex` as the type representing
   * `AbstractSimplex[Int]`
   *
@@ -25,11 +49,11 @@ import math.Ordering.Implicits.sortedSetOrdering
   *   and override everything to get the type to print out at `Simplex`
   *   everywhere instead of as the more verbose `AbstractSimplex`
   */
-type Simplex = AbstractSimplex[Int]
-
-object Simplex {
-  def apply(vertices: Int*) = new Simplex(TreeSet[Int](vertices: _*))
-}
+//type Simplex = AbstractSimplex[Int]
+//
+//object Simplex {
+//  def apply(vertices: Int*) = new Simplex(TreeSet[Int](vertices: _*))
+//}
 
 /** Class representing an abstract simplex. Abstract simplices are sets (of
   * totally ordered vertices) and thus are represented by a type that implements
@@ -139,3 +163,4 @@ object AbstractSimplex extends SortedIterableFactory[AbstractSimplex] {
     }
 
 }
+
diff --git a/src/main/scala/org/appliedtopology/tda4j/VietorisRips.scala b/src/main/scala/org/appliedtopology/tda4j/VietorisRips.scala
index 980bfb1c..72666036 100644
--- a/src/main/scala/org/appliedtopology/tda4j/VietorisRips.scala
+++ b/src/main/scala/org/appliedtopology/tda4j/VietorisRips.scala
@@ -1,10 +1,10 @@
 package org.appliedtopology.tda4j
 
+import org.appliedtopology.tda4j.FiniteMetricSpace.MaximumDistanceFiltrationValue
+
 import scala.collection.immutable.{LazyList, SortedSet}
 import scala.math.Ordering.Implicits.*
-import Simplex.*
-import org.appliedtopology.tda4j.FiniteMetricSpace.MaximumDistanceFiltrationValue
-import scalax.collection.{edge, mutable as gmutable, Graph}
+import scalax.collection.{Graph, edge, mutable as gmutable}
 import scalax.collection.edge.Implicits.*
 import scalax.collection.edge.WUnDiEdge
 
@@ -12,7 +12,7 @@ import scala.annotation.tailrec
 import scala.collection.mutable
 import scala.util.Sorting
 import scala.util.control.*
-import scala.util.chaining._
+import scala.util.chaining.*
 
 /** Convenience definition to allow us to choose a specific implementation.
   *
diff --git a/src/main/scala/org/appliedtopology/tda4j/package.scala b/src/main/scala/org/appliedtopology/tda4j/package.scala
index 78543ae6..d76e1b2d 100644
--- a/src/main/scala/org/appliedtopology/tda4j/package.scala
+++ b/src/main/scala/org/appliedtopology/tda4j/package.scala
@@ -1,5 +1,12 @@
 package org.appliedtopology
 
+import math.Ordering.Implicits.sortedSetOrdering
+
 /** Package for the Scala library TDA4j
   */
-package object tda4j {}
+package object tda4j {
+  class TDAContext[VertexT : Ordering, CoefficientT : Fractional]
+    extends ChainOps[AbstractSimplex[VertexT], CoefficientT], SimplexContext[VertexT] {
+    given Conversion[Simplex, Chain[Simplex,CoefficientT]] = Chain.apply
+  }
+}
diff --git a/src/test/scala/org/appliedtopology/tda4j/APISpec.scala b/src/test/scala/org/appliedtopology/tda4j/APISpec.scala
new file mode 100644
index 00000000..414eab4e
--- /dev/null
+++ b/src/test/scala/org/appliedtopology/tda4j/APISpec.scala
@@ -0,0 +1,19 @@
+package org.appliedtopology.tda4j
+
+import org.specs2.mutable
+
+class APISpec extends mutable.Specification {
+  """Test case class for developing the non-Scala facing API functionality
+    |and the non-expert API functionality""".stripMargin
+
+  given ctx : TDAContext[Char, Double]()
+  import ctx.{*,given}
+
+  "we should be able to create and compute with chains" >> {
+    1.0 *> s(1,2) - s(2,3) should beEqualTo(
+      Chain(
+        Simplex(1,2) -> 1.0,
+        Simplex(2,3) -> -1.0)
+    )
+  }
+}
diff --git a/src/test/scala/org/appliedtopology/tda4j/ChainSpec.scala b/src/test/scala/org/appliedtopology/tda4j/ChainSpec.scala
index aa355482..f4d5bedf 100644
--- a/src/test/scala/org/appliedtopology/tda4j/ChainSpec.scala
+++ b/src/test/scala/org/appliedtopology/tda4j/ChainSpec.scala
@@ -10,13 +10,19 @@ class ChainSpec extends mutable.Specification {
   """This is the specification for testing the Chain implementation.
     |""".stripMargin.txt
 
-  given Conversion[AbstractSimplex[Int], Chain[AbstractSimplex[Int], Double]] =
+  given sc: SimplexContext[Int]()
+  import sc.*
+
+  given Conversion[Simplex, Chain[Simplex, Double]] =
     Chain.apply
 
   given Fractional[Double] = math.Numeric.DoubleIsFractional
 
   given Ordering[Int] = math.Ordering.Int
 
+  given rm : RingModule[Chain[Simplex, Double], Double] = ChainOps()
+  import rm.*
+
   "The `Chain` type should" >> {
     val z1 = Chain(Simplex(1, 2, 3))
     val z2 =
@@ -61,7 +67,7 @@ class ChainSpec extends mutable.Specification {
       def e1 = {
         val chain = z1
         val expectedResult = Chain(Simplex(1, 2, 3))
-        val result = 2 *: chain
+        val result = 2 *> chain
 
         result must beEqualTo(expectedResult)
       }
@@ -116,4 +122,33 @@ class ChainSpec extends mutable.Specification {
 
   }
 
+  "The `Chain` type should be comfortable to write expressions with" >> {
+    val z1 = Chain(s(1,2,3))
+    val z2 : Chain[Simplex,Double] = s(1,2) - s(1,3) + s(2,3)
+    val z3 = 1.0 *> s(1, 2, 5)
+    val z4 : Chain[Simplex,Double] = Simplex(1, 4, 8) <* 1.0
+    val z5 : Chain[Simplex,Double] = -s(1,2) + s(1,4) - s(2,3)
+    val z6 : Chain[Simplex,Double] = s(1,2) + s(2,3) - s(1,3)
+    val z7 : Chain[Simplex,Double] = s(1,2) - s(1,3) + s(2,3) + 0.0 *> s(3,4)
+
+    z1 must haveClass[Chain[Simplex, Double]]
+    z2 must beEqualTo(z6)
+    z2 must beEqualTo(z7)
+
+    val expectedResult1 = Chain(
+      Simplex(1, 2, 3) -> 1.0,
+      Simplex(1, 2) -> 1.0,
+      Simplex(1, 3) -> -1.0,
+      Simplex(2, 3) -> 1.0
+    )
+    val expectedResult2 = Chain(
+      Simplex(1, 2) -> 0.0,
+      Simplex(1, 3) -> 0.0,
+      Simplex(1, 4) -> 0.0,
+      Simplex(2, 3) -> 0.0
+    )
+
+    z1 + z2 must beEqualTo(s(1,2,3) + s(1,2) - s(1,3) + s(2,3))
+    z2 - z6 must beEqualTo(summon[RingModule[Chain[Simplex,Double], Double]].zero)
+  }
 }
diff --git a/src/test/scala/org/appliedtopology/tda4j/FiniteFieldSpec.scala b/src/test/scala/org/appliedtopology/tda4j/FiniteFieldSpec.scala
index 40e47025..4dd37a15 100644
--- a/src/test/scala/org/appliedtopology/tda4j/FiniteFieldSpec.scala
+++ b/src/test/scala/org/appliedtopology/tda4j/FiniteFieldSpec.scala
@@ -30,9 +30,9 @@ class FiniteFieldSpec extends mutable.Specification {
 
     val rand = Random
     val x = Fp(rand.between(-100, 100))
-    var y = Fp(rand.between(-100, 100))
+    var y = Fp(rand.between(1, 100))
     val z = Fp(rand.between(-100, 100))
-    if y == Fp(0) then y = Fp(rand.between(1, 100))
+    //if y == Fp(0) then y = Fp(rand.between(1, 100))
     "all operations stay within -p/2, p/2" >> {
       // noinspection ScalaRedundantConversion
       eg((x * y).toInt must beBetween(-8, 8))
diff --git a/src/test/scala/org/appliedtopology/tda4j/RingModuleSpec.scala b/src/test/scala/org/appliedtopology/tda4j/RingModuleSpec.scala
new file mode 100644
index 00000000..2e76d211
--- /dev/null
+++ b/src/test/scala/org/appliedtopology/tda4j/RingModuleSpec.scala
@@ -0,0 +1,37 @@
+package org.appliedtopology.tda4j
+import org.specs2.mutable
+
+class RingModuleSpec extends mutable.Specification {
+  """This is a test module for developing the RingModule[M,R] interface
+    |and make sure that it does what we need it to do.
+    |""".stripMargin
+
+  object rm extends RingModule[(Int,Int), Int] {
+    val zero: (Int, Int) = (0, 0)
+
+    def plus(ls: (Int, Int), rs: (Int, Int)): (Int, Int) = (ls._1 + rs._1, ls._2 + rs._2)
+
+    override def negate(x: (Int, Int)): (Int, Int) = (-x._1, -x._2)
+
+    def scale(x: Int, y: (Int, Int)): (Int, Int) = (x * y._1, x * y._2)
+  }
+
+  given RingModule[(Int,Int), Int] = rm
+
+  "zero should exist" >> {
+    val v : (Int,Int) = rm.zero
+    v must be_==(rm.zero)
+  }
+
+  "addition should work" >> {
+    ((2,3) + (4,5)) should be_== (6,8)
+  }
+
+  "scalar multiplication should work" >> {
+    (2,3) <* 4 should be_== (8,12)
+  }
+
+  "scalar left-multiplication should work" >> {
+    (4 *> (2,3)) should be_== (8,12)
+  }
+}
diff --git a/src/test/scala/org/appliedtopology/tda4j/SimplexSpec.scala b/src/test/scala/org/appliedtopology/tda4j/SimplexSpec.scala
index 3a6a2166..1a68a1c7 100644
--- a/src/test/scala/org/appliedtopology/tda4j/SimplexSpec.scala
+++ b/src/test/scala/org/appliedtopology/tda4j/SimplexSpec.scala
@@ -18,6 +18,9 @@ A `Simplex` should
   be able to remove a vertex (and then not have that vertex) $eRemoveVertex
 """
 
+  given sc : SimplexContext[Int]()
+  import sc.Simplex
+
   def s: Simplex = Simplex(1, 2, 3)
 
   def sAdd4: Simplex = s.incl(4)
@@ -46,6 +49,9 @@ A `Simplex` should
 class SimplexTypeSpec extends mutable.Specification {
   "This is a specification for the type level interactions of the `Simplex` class".txt
 
+  given sc: SimplexContext[Int]()
+  import sc.Simplex
+
   "The `Simplex` type should" >> {
     val s = Simplex(1, 2, 3)
     val t = Simplex(2, 3, 4)
@@ -102,3 +108,15 @@ class SimplexTypeSpec extends mutable.Specification {
     }
   }
 }
+
+class SimplexContextSpec extends mutable.Specification {
+  """Testing the implicit context approach to working with simplices"""
+
+  given sc : SimplexContext[Char]()
+  import sc.*
+
+  "we can create a simplex, and get the right type" >> {
+    s('a','b','c') must haveClass[Simplex]
+    s('a','b','c') must haveClass[AbstractSimplex[Char]]
+  }
+}
\ No newline at end of file
diff --git a/src/test/scala/org/appliedtopology/tda4j/SimplexStreamSpec.scala b/src/test/scala/org/appliedtopology/tda4j/SimplexStreamSpec.scala
index 002b16a1..da10d7d7 100644
--- a/src/test/scala/org/appliedtopology/tda4j/SimplexStreamSpec.scala
+++ b/src/test/scala/org/appliedtopology/tda4j/SimplexStreamSpec.scala
@@ -8,6 +8,9 @@ import scala.collection.mutable as cmutable
 class SimplexStreamSpec extends mutable.Specification {
   "Test suite for the abstraction and implementations of `SimplexStream`".txt
 
+  given sc : SimplexContext[Int]()
+  import sc.*
+
   val esb: ExplicitStreamBuilder[Int, Double] =
     ExplicitStreamBuilder[Int, Double]()