doubly indexed variables in a non-commutative ring

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Symbolic variables don't have a notion of index directly, just a name. You can create a list of variables to send to the var() command that have a systematic naming scheme that corresponds to double indexing:

sage: var(','.join(['x_%d%d' % (i,j) for i in [1..6] for j in [1..6]]))
(x_11, x_12, x_13, x_14, x_15, x_16, x_21, x_22, x_23, x_24, x_25, x_26, x_31, x_32, x_33, x_34, x_35, x_36, x_41, x_42, x_43, x_44, x_45, x_46, x_51, x_52, x_53, x_54, x_55, x_56, x_61, x_62, x_63, x_64, x_65, x_66)

... and they even typeset correctly:

sage: latex(x_11)
x_{11}

@benjaminfjones: I think what the OP wanted was how to define this in case of PolynomialRing. Something similar to this (obtained from the documentation):

sage: PolynomialRing(GF(7), 'y', 5)
Multivariate Polynomial Ring in y0, y1, y2, y3, y4 over Finite Field of size 7

The answer is a simple modification to what you did, since I don't think two variable indices can be automatically created. But we can just create the same list as you did and provide it as the second argument:

sage: R = PolynomialRing(ZZ, ['x_%d%d' % (i,j) for i in [1..2] for j in [1..2]]); R
Multivariate Polynomial Ring in x_11, x_12, x_21, x_22 over Integer Ring
sage: R.inject_variables()
Defining x_11, x_12, x_21, x_22
sage: x_11*x_12 # Now we can use the variables
x_11*x_12